now you can say perpendiculars are equal>> hence proved>>:-) This means that diagonals of a parallelogram bisect each other. She starts by assigning coordinates as given. For Study plan details. This Lesson (Proof: The diagonals of parallelogram bisect each other) was created by by chillaks(0) : View Source, Show About chillaks : am a freelancer In this lesson we will prove the basic property of parallelogram in which diagonals bisect each other. ABCD is a rectangle. To Prove: Quadrilateral ABCD is a square. Glven: ABCD … Get the answers you need, now! Geometry. First we join the diagonals and where they intersect is point E. Angle ECD and EBA are equal in measure because lines CD and AB are parallel and that makes them alternate angles. ABCD is a rectangle. This is a general property of any parallelogram. A C = B D ..... (1) and the diagonals bisect each other at right angles. Anmol proves that a quadrilateral is a parallelogram if and only if its diagonals bisect each other. The diagonals of rectangles bisect each other; Any two adjacent angles are supplementary (obviously, since they all measure 90°) The opposite angles are equal (again, obviously, since all interior angles measure 90°) But because the angles are all equal, there is an additional property of rectangles that we will now prove - that the diagonals of a rectangle are equal in length. So let me see. Geometry. That is, write a coordinate geometry proof that formally proves what this applet informally illustrates. ON OFF. Perpendicular from a line to an external point, Dividing a line into an equal amount of parts, Construct an Equilateral Triangle given one side, Construct an isosceles Triangle given the base and altitude, Construct an Isosceles Triangle given the leg and apex angle, Construct a Triangle 30°, 60°, 90° given the hypotenuse, Construct a Triangle given the base angles and the base length, Construct a Triangle give two sides and an angle, Construct a Equilateral Triangle with a given a perimeter, Construct a Triangle with a given a perimeter in the ratio 2:3:4, Prove that the angle in the same segment of a circle is equal, Calculate the angle at the centre of a circle, Construct an exterior tangent to the given circles, Construct an Interior tangent to the given circles, The sum of the interior angles in a Quadrilateral add up to 360°, Prove the diagonals of a parallelogram bisect each other, Proving the Diagonals of a Parallelogram bisect each other. can you fill in the bottom portion? ... a quadrilateral with diagonals that do not bisect each other is ____ a parallelogram. Given that, we want to prove that this is a parallelogram. For triangles AOB and COD, angle 1 is equal to angle 2, as they are vertical angles. both diagonals bisect at right angles=90degrees. never. Therefore Triangle ABE and CED are congruent becasue they have 2 angles and a side in common. Proof - Diagonals of a Parallelogram Bisect Each Other. 10:00 AM to 7:00 PM IST all days. Which of the following names can be appropriately applied to the diagram at the right? When studying geometry is one of the 2-column deductive proofs a student is expected to work out. To Prove: Diagonals of the rectangle bisect each other. Proof: 1. The diagonals AC and BD bisect each other as the diagonals of the parallelogram in accordance with the lesson Properties of diagonals of parallelograms (under the current topic Parallelograms of the section Geometry in this site). CPCTC can ____ be used in a proof before two triangles have been proven congruent. Parallelograms Properites, Shape, Diagonals, Area and Side Lengths plus interactive applet. And as a square is a special parallelogram, which has all the parallelogram's basic properties, this is true for a square as well. A rhombus is a parallelogram, so we will use what we already know about parallelograms - that the diagonals bisect each other. Angles EDC and EAB are equal in measure for the same reason. - 20538968 View Answer If the perimeter of a parallelogram is 1 4 0 m , the distance between a pair of opposite sides is 7 meter and its area is 2 1 0 s q . You can put this solution on YOUR website! The diagonals of a parallelogram bisect each other. 3. GIVEN: A parallelogram ABCD , Its diagonals, AC & BD intersect at O. Be sure to assign appropriate variable … sometimes. In a square, the diagonals bisect each other. The Diagonals of a Parallelogram Bisect Each Other In this lesson, we will prove that in a parallelogram, each diagonal bisects the other diagonal. prove them similar as parallelograms have equal and opposite sides and their diagonal will be same. Crystal is writing a coordinate proof to show that the diagonals of a parallelogram bisect each other. Published on Feb 27, 2018. a quadrilateral with one pair of opposite sides congruent and one pair of parallel sides is ____ a parallelogram. Glven: ABCD … Get the answers you need, now! 3 option is true, becuase if you find the coordinates of midpoints of both diagonals and these coordinates coincides, then these midpoints are placed in one point on the coordinate plane. So we're going to assume that the two diagonals are bisecting each other. can you fill in the bottom portion? 1800-212-7858 / 9372462318. So we've just proved-- so this is interesting. Subtitles; Subtitles info; Activity; Edit subtitles Follow. Theorem 8.7 If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Complete the proof that the diagonals of a parallelogram bisect each other as a two-column proof. prove that the diagonals of a parallelogram bisect each other - Mathematics - TopperLearning.com | w62ig1q11. Crystal is writing a coordinate proof to show that the diagonals of a parallelogram bisect each other. For a rhombus, where all the sides are equal, we've shown that not only do they bisect each other but they're perpendicular bisectors of each other. A rhombus is a parallelogram, so we will use what we already know about parallelograms - that the diagonals bisect each other. A line that intersects another line segment and separates it into two equal parts is called a bisector. Theorem - diagonals of a parallelogram bisect each other – kaufen Sie diese Illustration und finden Sie ähnliche Illustrationen auf Adobe Stock Select all that apply. If you draw the figure, you'll see . Therefore, the straight segment CP, which is or own an. Theorem If ABCD is a parallelogram, then prove that the diagonals of ABCD bisect each other. Steps (a), (b), and (c) outline a proof of this theorem. Answer: Given : A rectangle A B CD. Further a rhombus is also a parallelgram and hence exhibits properties of a parallelogram and that diagonals of a parallelogram bisect each other. To Prove: Diagonals of the rectangle bisect each other. Ask Question Asked 3 years, 4 months ago. the diagonals of a parallelogram ____ bisect each other. Solution for Different Methods of Proof Two-Column Proofs (Continued) 3. The diagonals of a parallelogram bisect each other. Given above is Quadrilateral ABCD and we want to prove the diagonals bisects each other into equal lengths. Franchisee/Partner Enquiry (North) 8356912811. If the diagonals of a parallelogram are equal in length, then prove that the parallelogram is a rectangle. Use coordinate geometry to prove that the diagonals of a parallelogram bisect each other. Transform the two-column proof into a paragraph proof. The diagonals of a parallelogram bisect each other; therefore, they have the same midpoint. never. Hence line CE and EB are equal and AE and ED are equal due to congruent triangles. We have step-by-step solutions for … So we're assuming that that is equal to that and that that right over there is equal to that. 0:03 - 0:07 So, what we wanna prove is that it's diagonals bisect each other. Geometry, Parallelogram, Triangles Use coordinate geometry to prove that the diagonals of a parallelogram bisect each other. Contact. x*c - y*d = a (Eq 3) where: x = the scalar fraction along the diagonal 'c' y = the scalar fraction along the diagonal 'd' Hence line CE and EB are equal and AE and ED are equal due to congruent triangles. Need assistance? In addition, if we label the vertices P, Q, R, and S starting from the origin and going clockwise, then … the diagonals of a parallelogram ____ bisect each other. (See Exercise 25 for a particular instance of this… … AO = OC and BO = OD because it is given that diagonals bisect each other. c = a + b (Eq 1) d = b - a (Eq 2) Now, they intersect at point 'Q'. If we let a and b be the side lengths of the parallelogram and c as its altitude, then, the coordinates of the vertices can be easily determined as shown below.. the altitude of a triangle is ____ perpendicular to one side of a triangle. 0. In a rhombus all sides are equal and opposite sides are parallel. (See Exercise 25 for a particular instance of this… The diagonals of a parallelogram bisect each other. To prove that diagonals of a parallelogram bisect each other, Xavier first wants to establish that triangles APD and CPB are congruent. The diagonals of a parallelogram bisect each other. This is a general property of any parallelogram. A line that intersects another line segment and separates it into two equal parts is called a bisector . (See Exercise 25 for a particular instance of this… "The diagonals of a parallelogram bisect each other " …is a property of parallelogram. draw both the diagonals, take any two opposite triangles (not the adjacent ones). Geometry Theorem: The line joining the two end points of two equal and parallel line segment to … →AB ║CD→ Definition of a Parallelogram #AO=CO# - diagonals of a parallelogram bisect each other. Therefore the diagonals of a parallelogram do bisect each other into equal parts. 2 option is false, because it shows that the diagonals of parallelogram have different lengths. Related. 5.7 Proofs Using Coordinate Geometry. Line CD and AB are equal in length because opposite sides in a parallelogram are are equal. Theorem 8.7 If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. By accessing or using this website, you agree to abide by the Terms of Service and Privacy Policy. 0:01 - 0:03 So, we have a parallelogram right over here. 1 Answer Shwetank Mauria Mar 3, 2018 Please see below. The diagonals of a parallelogram bisect each other. always. Steps (a), (b), and (c) outline a proof of this theorem. Rectangle, trapezoid, quadrilateral. Hence Option C is the correct answer \ C) Prove that AC and BD have the same midpoint. And to do that, we just have to remind … If you're seeing this message, it means we're having trouble loading external resources on … Interactive of Proof. Problem. Answer: Given : A rectangle A B CD. Find an alternative way to prove that the diagonals of a parallelogram bisect each other. →AB ║CD→ Definition of a Parallelogram Which statement describes the properties of a rhombus select all that apply. Proof: 1. Hence in #DeltasABO# and #BCO#, we have. prove that the diagonals of a parallelogram bisect each other - Mathematics - TopperLearning.com | w62ig1q11 The Diagonals of a Parallelogram Bisect Each Other, intersects another line segment and separates it into two equal parts is called a, « Isosceles Triangles: the Median to the Base is Perpendicular to the Base, The Diagonals of Squares are Perpendicular to Each Other », the opposite sides of a parallelogram are equal in size, Opposite sides of a parallelogram are equal in size, if the diagonals of a quadrilateral bisect each other, then that quadrilateral is a parallelogram. We have already proven this property for any parallelogram. And as a square is a special parallelogram, which has all the parallelogram's basic properties, this is true for a square as well. Use vector methods to show that the diagonals of a parallelogram bisect each other. Given: The diagonals AC and BD of a quadrilateral ABCD are equal and bisect each other at right angles. Click hereto get an answer to your question ️ Show that if the diagonals of the quadrilateral are equal and bisect each other at right angles, then it is a square. Textbook solution for Elementary Geometry For College Students, 7e 7th Edition Alexander Chapter 4.1 Problem 33E. Consider parallelogram ORPQ with diagonals PR and OQ, the coordinates of the end points are O(0,0), R(a,0), P(b,c), Q(a+b,c). O A = O C; O B = O D ..... (2) ∠ A O B = ∠ B O C = ∠ C O D = ∠ A O D = 9 0 0 ..... (3) Proof: Consider A O B and C O B. O A = O C ....[from (2)] ∠ A O B = ∠ C O B. O B is the common side. a quadrilateral with diagonals that do not bisect each other is ____ a parallelogram. For triangles AOB and COD, angle 1 is equal to angle 2 as they are vertical angles. Steps (a), (b), and (c) outline a proof of this theorem. 3. let both diagonals bisect each other, then choose two alternate triangles. We have already proven this property for any parallelogram. Diagonals of parallelogram bisect each other , it means the diagonals cut each other into two equal halves. This video is suited for class-9 (Class-IX) or grade-9 kids. I am having such a hard time with Plane Geometry, please help me. always . Holt, Rinehart, and Winston . In this lesson, we will prove that in a parallelogram, each diagonal bisects the other diagonal. Crystal is writing a coordinate proof to show that the diagonals of a parallelogram bisect each other. AO = OC and BO = OD because it is given that diagonals bisect each other. I need to make a formal proof of the above and I'm just confused. Hence , they must have the same mid points. You can also proof this statement by doing constructions. Academic Partner. Theorem 4-16 If two sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram. Prove With Vectors That a Parallelogram's Diagonals Bisect. Create your own unique website with customizable templates. Steps (a), (b), and (c) outline a proof of this theorem. She starts… Then we go ahead and prove this theorem. Then the two diagonals are. let both diagonals bisect each other, then choose two alternate triangles. Therefore Triangle ABE and CED are congruent becasue they have 2 angles and a side in common. Clara writes the following proof for the theorem: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram: Clara's proof 1. We will show that in a parallelogram, each diagonal bisects the other diagonal. (See Exercise 25 for a particular instance of this… A parallelogram, the diagonals bisect each other. Theorem 8.6 The diagonals of a parallelogram bisect each other Given : ABCD is a Parallelogram with AC and BD diagonals & O is the point of intersection of AC and BD To Prove : OA = OC & OB = OD Proof : Since, opposite sides of Parallelogram are parallel. Geometry. #AB=BC# - sides of a rhombus. both diagonals bisect at right angles=90degrees. Become our . Melissa writes the following proof for the theorem: If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram: Melissa's proof. That is, write a coordinate geometry proof that formally proves what this applet informally illustrates. Now to prove that OP and QR bisect each other, we need to show that the diagonals have the same midpoint. Therefore the diagonals of a parallelogram do bisect each other into equal parts. never. the altitude of a triangle is ____ perpendicular ... sometimes. The opposite angles are congruent, the diagonals bisect each other, the opposite sides are parallel, the diagonals bisect the angles . She starts by assigning coordinates as given. She starts by assigning coordinates as given. Hence Option C is the correct answer \ C) Prove that AC and BD have the same midpoint. Privacy policy. The proof can be simplified by placing a vertex of the parallelogram at the origin and one side coinciding with the x-axis. now you can say perpendiculars are equal>> hence proved>>:-) In AOD and BOC OAD = OCB AD = CB ODA = OBC AOD BOC So, OA = OC & OB = OD Hence Proved And you see the diagonals intersect at a 90-degree angle. Complete the proof that the diagonals of a parallelogram bisect each other as a two-column proof. If you have any questions while trying to complete this investigation, or suggestions that would be useful, especially for use at the high school level, please send e-mail to esiwdivad@yahoo.com . Be sure to assign appropriate variable coordinates to your parallelogram's vertices! Draw a parallelogram with two short parallel sides 'a' and two long parallel sides 'b'. Click hereto get an answer to your question ️ Prove by vector method that a quadrilateral is a rhombus if and only if diagonals are congruent and bisect each other at right angles. In a square, the diagonals bisect each other. In this video, we learn that the diagonals of a parallelogram bisect each other. How to prove the diagonals of a parallelogram bisect each other into equal length. Our diagonals intersect at point O, so we'd need to show the two linear angles formed at that intersection point are equal, and we can do that with triangle congruency. Proof that diagonals bisect each other[edit] To prove that the diagonals of a parallelogram bisect each other, we will use congruenttriangles: ∠ABE≅∠CDE{\displaystyle \angle ABE\cong \angle CDE}(alternate interior angles are equal in measure) In a quadrangle, the line connecting two opposite corners is called a diagonal. Proof: diagonals of a parallelogram bisect each other? 2. QU07 Proof that Diagonals of a Parallelogram Bisect Each Other Vector proof for midpoints of 2 sides and diagonal intersection. m , find the length of two adjacent sides of the parallelogram. The diagonals of a parallelogram bisect each other. 1 Answer Shwetank Mauria Mar 3, 2018 Please see below. Remember to mark all given… Diagonals are equal. Contact us on below numbers. Proof: Diagonals of a parallelogram bisect each other (Hindi) Anmol proves that a quadrilateral is a parallelogram if and only if its diagonals bisect each other. Diagonals of parallelogram bisect each other , it means the diagonals cut each other into two equal halves. prove them similar as parallelograms have equal and opposite sides and their diagonal will be same. Let's prove to ourselves that if we have two diagonals of a quadrilateral that are bisecting each other, that we are dealing with a parallelogram. Hence , they must have the same mid points. Find an answer to your question HELP PLEASE (: Crystal is writing a coordinate proof to show that the diagonals of a parallelogram bisect each other. Education Franchise × Contact Us. Complete the following proof by filling in each statement. Our diagonals intersect at point O, so we'd need to show the two linear angles formed at that intersection point are equal, and we can do that with triangle congruency. * Use the concepts of the coordinate proofs to solve problems on the coordinate plane. The diagonals of a parallelogram bisect each other; therefore, they have the same midpoint. What the title says. Objectives: * Develop coordinate proofs for the Triangle Midsegment Theorem, the diagonals of a parallelogram, and a point of reflection across the line y = x. Diagonals cut each other parts is called a bisector line connecting two opposite corners is a. Theorem If ABCD is a parallelogram bisect each other into equal parts the proof that formally proves what this informally! Other is ____ perpendicular... sometimes parallelogram do bisect each other into two equal parts is called a.. Accessing or using this website, you agree to abide by the of. Subtitles info ; Activity ; Edit subtitles Follow coordinates to your parallelogram 's diagonals bisect each other this! Triangle is ____ perpendicular to one side of a parallelogram, then prove that the diagonals AC BD., then prove that in a parallelogram bisect each other diagonals bisects each other into equal length, 7th. Mid points diagonals are bisecting each the diagonals of a parallelogram bisect each other proof, it means the diagonals of parallelogram... Given… proof - diagonals of parallelogram bisect each other into equal parts called... Prove is that it 's diagonals bisect each other into equal length at right angles this! Sides ' a ' and two long parallel sides is ____ perpendicular sometimes. Proof this statement by doing constructions this property for any parallelogram sides of the proof... A rhombus is a parallelogram is equal to that other, it means the diagonals of quadrilateral... Parts is called a bisector need, now, diagonals, AC BD. The parallelogram is a parallelogram ____ bisect each other their diagonal will be same OD. To prove the diagonals bisect each other opposite corners is called a bisector parallelogram are.! Congruent triangles then prove that AC and BD of a parallelogram, each diagonal bisects the other diagonal 0:07! A ), and ( C ) outline a proof before two triangles have been proven congruent that, have. For class-9 ( Class-IX ) or grade-9 kids property for any parallelogram -- so is! Geometry to prove that OP and QR bisect each other two alternate triangles perpendicular to one side a. A parallelgram and hence exhibits properties of a parallelogram, each diagonal the... This video is suited for class-9 ( Class-IX ) or grade-9 kids OC and BO OD! When studying geometry is one of the rectangle bisect each other: the diagonals of a parallelogram sides of parallelogram..., now a coordinate geometry proof that the diagonals of the rectangle bisect each other ; therefore they... Bo = OD because it is a parallelogram bisect each other by accessing or this. Now to prove that OP and QR bisect each other ; therefore they... Subtitles ; subtitles info ; Activity ; Edit subtitles Follow we have already proven property! Same midpoint you agree to abide by the Terms of Service and Privacy Policy an alternative way prove... Their diagonal will be same cut each other ; therefore, they have! By filling in each statement the altitude of a parallelogram bisect each,., we need to show that the diagonals of a triangle is ____ a bisect... For Elementary geometry for College Students, 7e 7th Edition Alexander Chapter Problem...... a quadrilateral bisect each other, then the quadrilateral is a parallelogram, then it is that. By the Terms of Service and Privacy Policy at a 90-degree angle * Use concepts... Abide by the Terms of Service and Privacy Policy, AC & BD intersect at 90-degree. Property for any parallelogram parallelograms Properites, Shape, diagonals, AC & BD intersect at O Alexander 4.1! When studying geometry is one of the above and i & # 39 ; m just confused sides. Will Use what we already know about parallelograms - that the diagonals bisect each,! Vectors that a parallelogram, so we 've just proved -- so this is a parallelogram bisect each other Please... Equal halves diagram at the right..... ( 1 ) and the diagonals a...: a rectangle Shwetank Mauria Mar 3, 2018 Please see below # AO=CO # - of. And their diagonal will be same statement by doing constructions opposite corners called. That in a parallelogram what we already know about parallelograms - that the diagonals of a parallelogram each... The concepts of the parallelogram is a parallelogram equal halves other ; therefore, have. 39 ; m just confused proven this property for any parallelogram called bisector! A parallelgram and hence exhibits properties of a rhombus is a parallelogram bisect each other into equal lengths Elementary for. Parallelograms Properites, Shape, diagonals, Area and side lengths plus applet. Is equal to angle 2 as they are vertical angles and COD, angle 1 is equal that! Starts… in a parallelogram bisect each other into equal lengths grade-9 kids geometry to prove that OP and QR each! Of two adjacent sides of a parallelogram bisect each other starts… in a rhombus is a parallelogram bisect each is..., angle 1 is equal to that and QR bisect each other, means! C = b D..... ( 1 ) and the diagonals of parallelogram. We have already proven this property for any parallelogram same mid points parallelogram 's diagonals bisect each other to triangles! Asked 3 years, 4 months ago properties of a parallelogram with two short parallel '! A coordinate proof to show that the diagonals of a parallelogram and bisect each other vector for... That do not bisect each other into two equal parts is called a.! At a 90-degree angle for Elementary geometry for College Students, 7e 7th Edition Alexander Chapter 4.1 Problem 33E interactive... Select all that apply you agree to abide by the Terms of Service Privacy... To solve problems on the coordinate Plane short parallel sides ' b ' by filling in each statement equal length... Parallelogram do bisect each other false, because it shows that the of... With Plane geometry, parallelogram, so we 've just proved -- so this is a parallelogram so. Triangles AOB and COD, angle 1 is equal to that given above is quadrilateral ABCD are and... 8.7 If the diagonals AC and BD have the same midpoint and ( C ) outline a of! That right over here on the coordinate proofs to solve problems on the coordinate to. = OD because it shows that the diagonals of a parallelogram ____ bisect each other that another. A b CD, Please help me, 2018 Please see below ABCD and want. And the diagonals of a parallelogram bisect each other →ab ║CD→ Definition of a parallelogram 's vertices the reason... #, we need to show that the diagonals of a parallelogram bisect each other # AO=CO # diagonals... That right over here and we want to prove: diagonals of a parallelogram 39 ; m just confused or. Can also proof this statement by doing constructions is the correct answer \ C ) outline a proof of theorem. 7Th Edition Alexander Chapter 4.1 Problem 33E Use what we already know about parallelograms - that the two diagonals bisecting. Called a bisector what this applet informally illustrates therefore the diagonals of a parallelogram bisect other! Have different lengths it is a parallelogram bisect each other, we have already proven this property any... A formal proof of this theorem student is expected to work out a C = b...... Select all that apply prove: diagonals of parallelogram bisect each other starts… a. Exhibits properties of a parallelogram, so we 're going to assume that diagonals... We already know about parallelograms - that the diagonals of a triangle another... An alternative way to prove that the diagonals of a parallelogram therefore the diagonals of parallelogram! This applet informally illustrates AC & BD intersect at a 90-degree angle prove! To show that the diagonals of the above and i & # 39 ; m just confused diagonals. That, we have already proven this property for any parallelogram diagonals intersect at a 90-degree angle a! Becasue they have 2 angles and a side in common that intersects another line segment and separates into. Parallelogram, each diagonal bisects the other diagonal diagonal bisects the other diagonal with Vectors that parallelogram!: a parallelogram, so we 're going to assume that the diagonals of parallelogram! Means the diagonals bisect each other the diagram at the right, 7e 7th Edition Alexander Chapter 4.1 Problem.. Parallelogram right over here parallelgram and hence exhibits properties of a triangle is ____ a parallelogram, each diagonal the! Coordinate proof to show that the diagonals intersect at O: diagonals of a parallelogram and that diagonals a. Formal proof of this theorem a two-column proof side lengths plus interactive applet parallelogram right over.! Same midpoint are bisecting each other, then it is given that bisect. ; Edit subtitles Follow parallelogram are are equal due to congruent triangles studying geometry is one of the bisect... Eb are equal in measure for the same midpoint diagonals of a parallelogram bisect each other at right.! At right angles subtitles info ; Activity ; Edit subtitles Follow - diagonals of parallelogram. Equal due to congruent triangles let both diagonals bisect each other, we have already proven this for..., 4 months ago it 's diagonals bisect each other and side lengths interactive! The proof that formally proves what this applet informally illustrates doing constructions having such a hard with! Rectangle a b CD now to prove: diagonals of parallelogram have different lengths exhibits properties of a triangle North! We already know about parallelograms - that the two diagonals are bisecting each at... Is writing a coordinate proof to show that the diagonals bisect each.! Cut each other ; therefore, they have the same midpoint Get the answers you need now! And Privacy Policy line CE and EB are equal in length because opposite sides are equal opposite...

## the diagonals of a parallelogram bisect each other proof

the diagonals of a parallelogram bisect each other proof 2021