Geometry (Thailand Math POSN 1st elimination round 2014)

A lot easier than last year.

Write a full solution.

1.) Let $\square ABCD$ be a quadrilateral with the sum of the opposite sides are equal. (i.e. $AB + CD = AD + BC$). Prove that

• 1.1) $\square ABCD$ is a tangential quadrilateral (there is a circle inside the square and tangent to each sides).
• 1.2) The 4 internal angle bisectors intersect at one point.

2.) Let $\triangle ABC$ be the triangle with point $D$ lies on $\overline{BC}$ such that $AB\times DC = AC \times BD$. Prove that $\overline{AD}$ is the internal angle bisector of $B\hat{A}C$.

3.) Let $\triangle ABC$ be the triangle with $B\hat{A}C < 90^{\circ}$ and $BC = a, CA = b, AB = c$. If $\overline{AD}$ is a median line, prove that $\displaystyle 2(AD^{2}) = b^{2}+c^{2}-\frac{a^{2}}{2}$.

4.) Given a line segment length $1$ unit. Explain how to construct a square with area of $5\sqrt{3}$ sq.unit using only straightedge and compass.

5.) Let the external angle bisectors of $\triangle ABC$ bisect at the extension of sides of triangle at $D,E,F$. Prove that $D,E,F$ are collinear.

Check out all my notes and stuffs for more problems!

Thailand Math POSN 2013

Thailand Math POSN 2014