Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/242

218 PROPOSITION C. THEOREM.

If from the vertical angle of a triangle a straight line, be drawn perpendicular to the base, the rectangle contained by the sides of the triangle is equal to the rectangle con-tained by the perpendicular and the diameter of the circle described about the triangle.

Let ABC be a triangle, and let AD be the perpendicular from the angle A to the base BC the rectangle BA, AC shall be equal to the rectangle contained by AD and the diameter of the circle described about the triangle.

Describe the circle ACB about the triangle; [IV. 5. draw the diameter AE, and join EC

Then, because the right angle BDA is equal to the angle ECA in a semi-circle; [III. 31. and the angle ABD is equal to the angle AEC, for they are in the same segment of the circle; [III. 21. therefore the triangle ABD is equiangular to the triangle AEC. Therefore BA is to AD as EA is to AC; [VI; 4. therefore the rectangle BA, AC is equal to the rectangle EA,AD. [ VI. 16.

Wherefore, if from the vertical angle &c.

PROPOSITION D. THEOREM.

The rectangle contained by the diagonals of a quadrilateral figure inscribed in a circle is equal to both the rectangles contained by its opposite sides.