Or again: Let O be the centre (fig. 2). Join OA, OB OC, OD. Then the four triangles AOB, BOC, COD, DOA are each isosceles. Hence the angle OAB is equal to the angle OBA, and the angle OAD equal to the angle ODA; therefore the angle BAD is equal to the sum of the angles OBA, ODA. In like manner the angle BCD is equal to the sum of the angles OBC, ODC. Hence the sum of the two angles BAD, BCD is equal to the sum of the two angles ABC, ADC, and hence each sum is two right angles. Cor.—If a parallelogram be inscribed in a circle it is a rectangle. Exercises. 1. If the opposite angles of a quadrilateral be supplemental, it is cyclic. 2. If a figure of six sides be inscribed in a circle, the sum of any three alternate angles is four right angles. 3. A line which makes equal angles with one pair of opposite sides of a cyclic quadrilateral, makes equal angles with the remaining pair and with the diagonals. 4. If two opposite sides of a cyclic quadrilateral be produced to meet, and a perpendicular be let fall on the bisector of the angle between them from the point of intersection of the diagonals, this perpendicular will bisect the angle between the diagonals. 5. If two pairs of opposite sides of a cyclic hexagon be respectively parallel to each other, the remaining pair of sides are also parallel. 6. If two circles intersect in the points A, B, and any two lines ACD, BFE, be drawn through A and B, cutting one of the circles in the points C, E, and the other in the points D, F, the line CE is parallel to DF. 7. If equilateral triangles be described on the sides of any triangle, the lines joining the vertices of the original triangle to the opposite vertices of the equilateral triangles are concurrent. 8. In the same case prove that the centres of the circles described about the equilateral triangles form another equilateral triangle. 9. If a quadrilateral be described about a circle, the angles at the centre subtended by the opposite sides are supplemental. 10. The perpendiculars of a triangle are concurrent. 11. If a variable tangent meets two parallel tangents it subtends a right angle at the centre. 12. The feet of the perpendiculars let fall on the sides of a triangle from any point in the circumference of the circumscribed circle are collinear (Simson). Def.—The line of collinearity is called Simson’s line. 13. If a hexagon be circumscribed about a circle, the sum of the angles subtended at the centre by any three alternate sides is equal to two right angles. PROP. XXIII—Theorem. Two similar segments of circles which do not coincide cannot be constructed on the same chord (AB), and on the same side of that chord.