MathDB

P9

Part of 2023 Bangladesh Mathematical Olympiad

Problems(2)

Big regular polygons

Source: BdMO 2023 Secondary National P9

2/12/2023
Let A1A2…A2nA_1A_2\dots A_{2n} be a regular 2n2n-gon inscribed in circle ω\omega. Let PP be any point on the circle ω\omega. Let H1,H2,…,HnH_1,H_2,\dots, H_n be orthocenters of triangles PA1A2,PA3A4,…,PA2n−1A2nPA_1A_2, PA_3A_4,\dots, PA_{2n-1}A_{2n} respectively. Prove that H1H2…HnH_1H_2\dots H_n is a regular nn-gon.
geometryorthocenterregular polygon
Cute Geometry Problem

Source: BdMO 2023 Higher Secondary National P9

2/12/2023
Let ΔABC\Delta ABC be an acute angled triangle. DD is a point on side BCBC such that ADAD bisects angle ∠BAC\angle BAC. A line ll is tangent to the circumcircles of triangles ADBADB and ADCADC at point KK and LL, respectively. Let MM, NN and PP be its midpoints of BDBD, DCDC and KLKL, respectively. Prove that ll is tangent to the circumcircle of ΔMNP\Delta MNP.
geometryparallelcyclic quadrilateralradical axisAngle ChasingInversion