Problem 1:

A triangle is divided into two parts by drawing a line through the centroid. Prove that the area of the smaller part is at least 80 % of the bigger part. In fact, this statement is true for all convex figures and is known as Winternitz theorem.

Problem 2:

In a rectangle ABCD, the side . Locate *geometrically *(use of only a compass and an unmarked straightedge is allowed) the points X and Y on CD, so that .

Problem 3:

In a triangle ABC, . Locate *geometrically *the points D on AB and E on AC, so that .

Problem 4:

P is a point inside a square ABCD such that . Prove that the triangle PAB is equilateral.

Problem 5:

Four circles are drawn, all of same radius r and passing through a point O. Let the quadrilateral ABCD consisting of direct tangents to this set of circles be the circumscribing quadrilateral. Prove that ABCD is a cyclic quadrilateral.

Problem 6:

**Viviani’s Theorem: **Prove that the sum of the distances of a point inside an equilateral triangle from the three sides is independent of the point.

Problem 7:

Geometrically construct a lune (a concave area bounded by two circular arcs) of unit area.

*Have fun!*

Nalin Pithwa.

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