MathDB

3

Part of 2017 Irish Math Olympiad

Problems(2)

A'B'C'D' is similar to ABCD

Source: Irish MO 2017 paper 1 problem 3

12/12/2022
Four circles are drawn with the sides of quadrilateral ABCDABCD as diameters. The two circles passing through AA meet again at A′A', two circles through BB at B′B' , two circles at CC at C′C' and the two circles at DD at D′D'. Suppose the points A′,B′,C′A',B',C' and D′D' are distinct. Prove quadrilateral A′B′C′D′A'B'C'D' is similar to ABCDABCD.
similaritygeometrysimilar
\sum |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2

Source: Irish MO 2017 paper 2 problem 3

12/12/2022
A line segment B0BnB_0B_n is divided into nn equal parts at points B1,B2,...,Bn−1B_1,B_2,...,B_{n-1} and AA is a point such that ∠B0ABn\angle B_0AB_n is a right angle. Prove that : ∑k=0n∣ABk∣2=∑k=0n∣B0Bk∣2\sum_{k=0}^{n} |AB_k|^{2} = \sum_{k=0}^{n} |B_0B_k|^2
geometrySum