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Pranav
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 Nov 4, 2013
 428
Problem:
Given the base of a triangle and sum of its sides, then the locus of the centre of its incircle is:
A)straight line
B)circle
C)ellipse
D)hyperbola
Attempt:
Since the given answer is ellipse, I tried showing that the sum of the distances of incentre from the end points of base is constant.
The side lengths opposite to vertex A, B and C are a, b and c respectively. As per the question, a (base) and a+b+c is given. The distance IB is $r/\sin(B/2)$ and IC is $r/\sin(C/2)$, where $r$ is the radius of incircle. Hence,
$$IB+IC=r\left(\frac{1}{\sin(B/2)}+\frac{1}{\sin(C/2)}\right)\,\,\,\, (*)$$
I use the following formulas:
$$r=\frac{\Delta}{s}=\sqrt{\frac{(sa)(sb)(sc)}{s}}$$
$$\sin\left(\frac{B}{2}\right)=\sqrt{\frac{(sa)(sc)}{ac}}$$
$$\sin\left(\frac{C}{2}\right)=\sqrt{\frac{(sa)(sb)}{ab}}$$
Substituting in (*) and simplifying,
$$IB+IC=\sqrt{\frac{a}{s}}\left(\sqrt{c(sb)}+\sqrt{b(sc)}\right)$$
The terms outside the parentheses is constant, I am unable to prove that the terms inside the parentheses are a constant too.
Any help is appreciated. Thanks!
Given the base of a triangle and sum of its sides, then the locus of the centre of its incircle is:
A)straight line
B)circle
C)ellipse
D)hyperbola
Attempt:
Since the given answer is ellipse, I tried showing that the sum of the distances of incentre from the end points of base is constant.
The side lengths opposite to vertex A, B and C are a, b and c respectively. As per the question, a (base) and a+b+c is given. The distance IB is $r/\sin(B/2)$ and IC is $r/\sin(C/2)$, where $r$ is the radius of incircle. Hence,
$$IB+IC=r\left(\frac{1}{\sin(B/2)}+\frac{1}{\sin(C/2)}\right)\,\,\,\, (*)$$
I use the following formulas:
$$r=\frac{\Delta}{s}=\sqrt{\frac{(sa)(sb)(sc)}{s}}$$
$$\sin\left(\frac{B}{2}\right)=\sqrt{\frac{(sa)(sc)}{ac}}$$
$$\sin\left(\frac{C}{2}\right)=\sqrt{\frac{(sa)(sb)}{ab}}$$
Substituting in (*) and simplifying,
$$IB+IC=\sqrt{\frac{a}{s}}\left(\sqrt{c(sb)}+\sqrt{b(sc)}\right)$$
The terms outside the parentheses is constant, I am unable to prove that the terms inside the parentheses are a constant too.
Any help is appreciated. Thanks!
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