More problems in pure plane geometry for RMO

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 AB>BC. Locate geometrically (use of only a compass and an unmarked straightedge is allowed) the points X and Y on CD, so that AX=XY=YB.

Problem 3:

In a triangle ABC, AB/2<AC<AB. Locate geometrically the points D on AB and E on AC, so that BD=DE=EC.

Problem 4:

P is a point inside a square ABCD such that \angle{PCD}=\angle{PDC}=15 degrees. 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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