Construct triangle, incircle and ninepoint circle given

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Construct triangle, incircle and ninepoint circle given

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Source: India tst 2002 p7

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Sayan

#1 h Jul 13, 2012, 3:52 AM • 2 Y

Y by Adventure10, Mango247

Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.

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#2 h Jul 30, 2012, 5:15 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Sayan wrote:

Given two distinct circles touching each other internally, show how to construct a triangle with the inner circle as its incircle and the outer circle as its nine point circle.

Let the two circles be $\omega, \gamma$ with $\omega$ being larger. Let the centres of $\omega, \gamma$ be $I,N$ respectively. Let $F$ be the point of tangency.

Now, let the triangle (which we have not constructed yet) be $\triangle ABC$ , with points of tangency of $CA,AB$ with incircle being $Q,R$ respectively. Let midpoint of $AB$ be $X$ .
We will construct $\triangle ABC$ with the following properties, and show that it is a required triangle:
1. $\angle BAC = 60^{\circ}$ .
2. $I$ is the incentre of $\triangle ABC$ .
3. $X$ lies on $\omega$ .
4. $F$ lies on $AI$ .
To construct this, just take an arc $QR$ of $\gamma$ , with midpoint of the minor arc $QR$ being $F$ and $\angle QIR = 120^{\circ}$ . Now, take $A$ to be the point of intersection of the tangents to $\gamma$ at $Q,R$ . Now, just take one of the intersections of $AQ$ with $\omega$ to be $X$ , and take $B$ so that $X$ is midpoint of $AB$ . Take $C$ to be the intersection of the second tangent to $\gamma$ from $B$ with $AR$ . It is easy to see that all 4 properties are true.

Now, let us prove that $\omega$ is the nine point circle of $\triangle ABC$ . To see this, note that that if we're given two points and a circle passing through exactly one of them, then there exists exactly one circle passing through both points and tangent to the given circle. Applying this here, we get that there is exactly one circle passing through $X,F$ tangent to $\gamma$ . It is well known that the nine point circle of $\triangle ABC$ satisfies this(For example, see this). Now, as $\omega$ also satisfies this, therefore, the nine point circle must be $\omega$ .

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