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.

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## Comments

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

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

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