1°. Rhombus. 2°. A mean proportional. 3°. Similar triangles. 4°. A segment of a circle. 5°. The apothem of a regular polygon.
Theorem: The perpendicular is the shortest line between a point and a straight line.
Theorem: In the same circle or equal circles, the less of two chords is at the greater distance from the centre; conversely, the chord at the greater distance from the centre is the less.
Construction: Divide a given straight line internally in extreme and mean ratio.
Theorem: The areas of two triangles which have an angle of one equal to the angle of the other are to each other as the products of the sides including those angles.
Problem: Given a circle of unit diameter and the side of a regular inscribed polygon, find the side of a regular inscribed polygon of double the number of sides.
Theorem: The four bisectors of the four angles of a quadrilateral form a second quadrilateral, the opposite angles of which are supplementary.
Theorem: If on the diameter of a circle two points be taken equally distant from the centre, the sum of the squares of the distances of any point of the circumference from these two points is constant.
Problem: Find the locus of the point of intersection of the three altitudes of a triangle, given a fixed base, and constant angle at the vertex.