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# Geometry (Thailand Math POSN 2nd round)

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1. Let $$H$$ be orthocenter of $$\triangle ABC$$ and point $$I,J,K$$ be a midpoint between each vertex $$A,B,C$$ and $$H$$ respectively. If $$P$$ is a point on circumcircle of $$\triangle ABC$$ not on the vertex $$A,B,C$$, and $$M$$ be the midpoint between $$P,H$$. Prove that $$I,J,K,M$$ lie on the same circle.

2. Let the tangents of circumcircle of $$\triangle ABC$$ at point $$B,C$$ intersect at point $$D$$. Prove that $$\overline{AD}$$ is symmedian line of $$ABC$$.

3. Let $$P,Q$$ be 2 points that are isogonal conjugate to each other in $$\triangle ABC$$. If $$\overline{PP_{1}}$$ and $$\overline{QQ_{1}}$$ is perpendicular to $$\overline{BC}$$ at point $$P_{1}$$ and $$Q_{1}$$ respectively, $$\overline{PP_{2}}$$ and $$\overline{QQ_{2}}$$ is perpendicular to $$\overline{CA}$$ at point $$P_{2}$$ and $$Q_{2}$$ respectively, and $$\overline{PP_{3}}$$ and $$\overline{QQ_{3}}$$ is perpendicular to $$\overline{AB}$$ at point $$P_{3}$$ and $$Q_{3}$$ respectively. Prove that $$P_{1},P_{2},P_{3},Q_{1},Q_{2},Q_{3}$$ lie on the same circle, and the center of that circle lies in the midpoint of $$P$$ and $$Q$$.

4. Let $$I,N,H$$ be the center of incircle, nine-point circle, and orthocenter of $$\triangle ABC$$ respectively. Construct $$\overline{ID}, \overline{NM}$$ perpendicular to $$\overline{BC}$$ at point $$D,M$$ respectively. If $$\overline{AH}$$ intersect circumcircle of $$\triangle ABC$$ at point $$K$$ such that $$Y$$ is a midpoint of $$\overline{AK}$$. Prove that $$|ID - NM| = \displaystyle \left |r - \displaystyle \frac{AY}{2}\right|$$ where $$r$$ is an inradius of $$\triangle ABC$$.

5. Let $$U$$ be a foot of altitude of $$\triangle ABC$$ from point $$A$$. If $$U',U''$$ are reflection of $$U$$ by $$\overline{CA},\overline{AB}$$ respectively, such that $$\overline{U'U''}$$ intersect $$\overline{CA},\overline{AB}$$ at point $$V,W$$ respectively. Prove that $$\overline{BV},\overline{CW}$$ is perpendicular to $$\overline{CA},\overline{AB}$$ respectively.

This note is a part of Thailand Math POSN 2nd round 2015.

Note by Samuraiwarm Tsunayoshi
2 years, 3 months ago

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The first question can be solved using homothety. The points A,B,C and P lie on the circumcircle. So consider a homothetic transformation of the circumcircle about orthocenter and shrink the circle to half of its radius. The mid points of H and A,B,C, P will lie on this circle. This is the nine point circle of the triangle. · 1 year, 7 months ago