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.
Dem.—If possible, let ACB, ADB, be two similar segments constructed on the same side of AB. Take any point D in the inner one. Join AD, and produce it to meet the outer one in C. Join BC, BD. Then since the segments are similar, the angle ADB is equal to ACB (Def. x.), which is impossible [I. xvi.]. Hence two similar segments not coinciding cannot be described on the same chord and on the same side of it.
PROP. XXIV.—Theorem.
Similar segments of circles (AEB, CFD) on equal chords (AB, CD) are equal to one another.
Dem.—Since the lines are equal, if AB be applied to CD, so that the point A will coincide with C, and the line AB with CD, the point B shall coincide with D; and because the segments are similar, they must coincide [xxiii.]. Hence they are equal.