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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
3 years, 2 months ago

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Comment deleted Nov 21, 2014

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Actually it's to prove the inverse theorem of angle bisector.

Samuraiwarm Tsunayoshi - 3 years, 2 months ago

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For q3 will you get marks if you quote Apollonius' Theorem? It is quite well known.

Joel Tan - 3 years, 2 months ago

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Nah, but it's actually really easy to prove.

Samuraiwarm Tsunayoshi - 3 years, 2 months ago

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