S L Loney Solutions for Chapter: Properties of Triangle, Exercise 3: Examples XXXVII

Through the angular points of a triangle are drawn straight line which make the same angle $\alpha$ with the area of opposite sides of the triangle; Prove that the area of the triangle formed by them is to the original triangle as $4{\cos }^2\alpha :1$.

Two circles, of radii $a$ and $b$, cut each other at an angle $\theta$. Prove that the length of the common chord is $\frac{2ab\sin \theta }{\sqrt{a^2+b^2+2ab\cos \theta }}$.

Three equal circles touch one another; find the radius of the circles which touches all three.

Three circles, whose radii are $a, b$ and $c$, touch one another externally and the tangents at their points of contact meet in a point; prove that the distance of this point from either of their points of contact is ${\left(\frac{abc}{a+b+c}\right)}^{\frac{1}{2}}$.

In triangle $ABC$ in the sides $BC, CA, AB$ are taken three points $A^{\text{'}}, B^{\text{'}}, C^{\text{'}}$ such that $B{A}^{\text{'}}:A^{\text{'}}C=C{B}^{\text{'}}:B^{\text{'}}A=A{C}^{\text{'}}:C^{\text{'}}B=m:n$;

Prove that if $A{A}^{\text{'}}, B{B}^{\text{'}}$ and $C{C}^{\text{'}}$ be joined they will form by their intersections a triangle whose area is to that of the triangle $ABC$ as ${\left(m-n\right)}^2:m^2+mn+n^2$.

The circle inscribed in the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in the points $A_1, B_1$ and ${\mathrm{C}}_1$, respectively. Similarly, the circle inscribed in the triangle $A_1{B}_1{C}_1$ touches the sides in $A_2, B_2, C_2$, respectively and so on; if $A_n{B}_n{C}_n$ be the $n^{th}$ triangle so formed, prove that its angles are $\frac{\mathrm{\pi }}{3}+{\left(-2\right)}^{-n}\left(A-\frac{\mathrm{\pi }}{3}\right), \frac{\mathrm{\pi }}{3}+{\left(-2\right)}^{-n}\left(B-\frac{\mathrm{\pi }}{3}\right)$ and $\frac{\mathrm{\pi }}{3}+{\left(-2\right)}^{-n}\left(C-\frac{\mathrm{\pi }}{3}\right)$.

Hence prove that the triangle so formed is ultimately equilateral.

$A_1{B}_1{C}_1$ is the triangle formed by joining the feet of the perpendiculars drawn from $ABC$ upon the opposite sides; in like manner $A_2{B}_2{C}_2$ is the triangle obtained by joining the feet of the perpendiculars from $A_1, B_1$ and $C_1$ on the opposite sides and so on. Find the values on the angles $A_n, B_n$ and $C_n$ in the $n^{th}$ of these triangles.

The legs of a tripod are each $10 \mathrm{c}\mathrm{m}$ in length and their points of contact with a horizontal table which the tripod stands from a triangle whose sides are $\mathrm{7,} 8$ and $9 \mathrm{c}\mathrm{m}$ in length. Find the inclination of the legs to the horizontal and the height of the apex.