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# 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

Note by Samuraiwarm Tsunayoshi
2 years, 5 months ago

## Comments

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Comment deleted Nov 21, 2014

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Actually it's to prove the inverse theorem of angle bisector. · 2 years, 4 months ago

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For q3 will you get marks if you quote Apollonius' Theorem? It is quite well known. · 2 years, 4 months ago

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Nah, but it's actually really easy to prove. · 2 years, 4 months ago

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