Elementary Geometry

Miscellaneous Propositions

947

Definition

A centre of similarity of two plane curves is a point such that any straight line being drawn through it to cut the curves, the segments of the line intercepted between the point and the curves are in a constant ratio. 948 If 𝐴𝐵, 𝐴𝐶 touch a circle at 𝐵 and 𝐶, then any straight line 𝐴𝐸𝐷𝐹, cutting the circle, is divided harmonically by the circumference and the chord of contact 𝐵𝐶.

Proof

from 𝐴𝐸⋅𝐴𝐹=𝐴𝐵²III.36 𝐴𝐵²=𝐵𝐷⋅𝐷𝐶+𝐴𝐷²923 and 𝐵𝐷⋅𝐷𝐶=𝐸𝐷⋅𝐷𝐹III.35 949 If 𝛼, 𝛽, 𝛾, in the same figure, be the perpendiculars to the sides of 𝐴𝐵𝐶 from any point 𝐸 on the circumference of the circle, then 𝛽𝛾=𝛼²

Proof

Draw the diameter 𝐵𝐻=𝑑; then 𝐸𝐵²=𝛽𝑑, because 𝐵𝐸𝐻 is a right angle. Similarly 𝐸𝐶²=𝛾𝑑. But 𝐸𝐵⋅𝐸𝐶=𝛼𝑑 (VI. D), therefore ⋯ 950 If 𝐹𝐸 be drawn parallel to the base 𝐵𝐶 of a triangle, and if 𝐸𝐵, 𝐹𝐶 intersect in 𝑂, then 𝐴𝐸∶𝐴𝐶∷𝐸𝑂∶𝑂𝐵∷𝐹𝑂∶𝑂𝐶 BY VI.2, Since each ratio=𝐹𝐸∶𝐵𝐶,
Cor: If 𝐴𝐶=𝑛⋅𝐴𝐸, then 𝐵𝐸=(𝑛+1)𝑂𝐸 951 The three lines drawn from the angles of a triangle to the middle point of the opposite sides, intersect in the same point, and divide each other in the ratio of two to one.
For, by the last theorem, any one of these lines is divided by each of the others in the ratio of two to one, measuring from the same extremity, and must therefore be intersected by them in the same point.
This point will be referred to as the centroid of tje triangle. 952 The perpendiculars from the angles upon the opposite sides of a triangle intersect in the same point.
Draw 𝐵𝐸, 𝐶𝐹 perpendicular to the sides, and let them intersect in 𝑂. Let 𝐴𝑂 meet 𝐵𝐶 in 𝐷. Circles will circumscribe 𝐴𝐸𝑂𝐹 and 𝐵𝐹𝐸𝐶, by (III.31);
therefore ∠𝐹𝐴𝑂=𝐹𝐸𝐵=𝐹𝐶𝐵III.21 therefore ∠𝐵𝐷𝐴=𝐵𝐹𝐶=a right angle; i.e. 𝐴𝑂 is perpendicular to 𝐵𝐶, and therefore the perpendicular from 𝐴 on 𝐵𝐶 passes through 𝑂.
𝑂 is called the orthocentre of the triangle 𝐴𝐵𝐶.

Cor

The perpendiculars on the sides bisect the angles of the triangle 𝐷𝐸𝐹, and the point 𝑂 is therefore the centre of the inscribed circle of that triangle.
Proof. From (III.21), and the circles circumscribing 𝑂𝐸𝐴𝐹 and 𝑂𝐸𝐶𝐷

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive