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Da Vinci's Proof
Transcript of Da Vinci's Proof
Prove that O is the center of the square ACJI, so that line BH goes through it.
Given: Triangle ABC is a right triangle, line HI is congruent to BC, HJ is congruent to BA, and O is the midpoint of line HB
1st.) Expand line BC and HJ until they meet at point K.
2ND.) Expand line BA and IH until they meet at point L.This makes square BKHL, now draw line IC.
3rd.) Prove triangles IOH and COB are congruent with ASA. A:Angles IBH and COB are congruent by vertical angles. S: Line BO is equal to HO because O is the midpoint of BH. A:Angles OBC and OHI are equal by alternate interior angles with lines BK and LH. SO line IO is equal to OC and O is the center of square ACJI.
Finding the Quadrilaterals
We have to prove next that angle DBG is a straight angle. We know that the diagonals of a square are angle bisectors, so angle DBE(45)+ angle EBF(90)+ angle FBG(45)=180
Next we need to prove that the quadrilaterals ABHI, JHBC,EDGF, and ADGC are all congruent to each other.
To prove the quadrilaterals, you require three angles and three sides. Sides HI = BC = CG = FG, AD = ED =AB = HJ, AI= CJ = EF = AC. Angles IHO = CBO = CGB = FGB, GFE = GCA = HIA = BCJ, FEB = CDA = CJH = IAB, ABH = JHB = ADG = EDG. We know this because of the sides of the squares and the congruent triangles, which we'll prove next.
Leonardo Da Vinci
Da Vinci's Proof
Lived from 1452 to 1519
Born and lived in Florence, Italy
Best known as an artist (
The Last Supper, Mona Lisa
Also made many scientific progress
Appeared in Assassin Creed
Assassin's Creed Davinci
Continuing Proof Again
Triangle JHI = Triangle ABC by SSS, S:HJ = AB, S:BC = HI, S:AC = JI. Then we know that Triangle ABC is congruent to Triangle EBF by SAS, S:AB = EB, A:ABC = EBF, S:BC = BF
The Key Idea
Because of the knowledge of the quadrilaterals being congruent then the hexagons are congruent. One hexagon has Squares A and B and two congruent triangles. The next hexagon has Square C and two congruent triangles. So the sum of the Squares A and B is equal to Square C, proving the Pythagorean Theorem.