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.
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 .
In a triangle ABC, . Locate geometrically the points D on AB and E on AC, so that .
P is a point inside a square ABCD such that . Prove that the triangle PAB is equilateral.
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.
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.
Geometrically construct a lune (a concave area bounded by two circular arcs) of unit area.