PROPOSITION 8. THEOREM.
In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.
Let ABC be a right-angled triangle, having the right angle BAC; and from the point A, let AD be drawn perpendicular to the base BC: the triangles DBA, DAC shall be similar to the whole triangle ABC, and to one another.
For, the angle BAC is equal to the angle BDA, each of them being a right angle, [Axiom 11. and the angle at B is common to the two triangles ABC, DBA;
therefore the remaining angle ACB is equal to the remaining angle DAB.
Therefore the triangle ABC is equiangular to the triangle DBA, and the sides about their equal angles are proportionals; [VI. 4.
therefore the triangles are similar. [VI. Definition 1.
In the same manner it may be shewn that the triangle DAC is similar to the triangle ABC.
And the triangles DBA, DAC being both similar to the triangle ABC, are similar to each other.
Wherefore, in a right-angled triangle &c. q.e.d.
Corollary. From this it is manifest, that the perpendicular drawn from the right angle of a right-angled triangle to the base, is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side.
For, in the triangles DBA, DAC,
BD is to DA as DA is to DC; [VI. 4.
and in the triangles ABC, DBA,
BC is to BA as BA is to BD; [VI. 4.
and in the triangles ABC,DAC,
BC is to CA as CA is to CD. [VI. 4.
