Geometry (Thailand Math POSN 1st elimination round 2014)

A lot easier than last year.

Write a full solution.

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

  • 1.1) ABCD\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 ABC\triangle ABC be the triangle with point DD lies on BC\overline{BC} such that AB×DC=AC×BDAB\times DC = AC \times BD. Prove that AD\overline{AD} is the internal angle bisector of BA^CB\hat{A}C.

3.) Let ABC\triangle ABC be the triangle with BA^C<90B\hat{A}C < 90^{\circ} and BC=a,CA=b,AB=cBC = a, CA = b, AB = c. If AD\overline{AD} is a median line, prove that 2(AD2)=b2+c2a22\displaystyle 2(AD^{2}) = b^{2}+c^{2}-\frac{a^{2}}{2}.

4.) Given a line segment length 11 unit. Explain how to construct a square with area of 535\sqrt{3} sq.unit using only straightedge and compass.

5.) Let the external angle bisectors of ABC\triangle ABC bisect at the extension of sides of triangle at D,E,FD,E,F. Prove that D,E,FD,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
4 years, 7 months ago

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

Joel Tan - 4 years, 7 months ago

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

Samuraiwarm Tsunayoshi - 4 years, 7 months ago

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