As shown in figure, from a point $P$ exterior of circle $\odot O$, we draw tangent $PA$ and the secant $PBC$. Let $AD \perp PO$ Prove that $AC$ is tangent to the circumcircle of $\vartriangle ABD$.
Problem
Source: China Northern MO 2011 p6 CNMO
Tags: geometry, tangent
JSJ20142014
20.03.2023 08:30
markmark
sunken rock
20.03.2023 10:32
Take $E$ reflection of $A$ about $PO, ABEC$ is then harmonic, $D$ is the midpoint of a diagonal, $AD$ is bisector of $\widehat{BDC}=2\widehat{BAC}$, so $\widehat{BDE}=\widehat{BAC}$, but $\widehat{BDE}=\widehat{BAD}+\widehat{ABD}$, while $\widehat{BAC}=\widehat{BAD}+\widehat{CAD}$, done. Best regards, sunken rock
trinhquockhanh
20.03.2023 11:03