# Incircle tangents in a triangle

• soopo
In summary: It has the three vertices of the triangle labeled C,B, and A. The ray from C to the point of tangency on the opposite side is labeled 1, the ray from B to the point of tangency on the opposite side is labeled 2, and the ray from A to the point of tangency on the opposite side is labeled 3. The line from C to A is also the bisector of the angle between 1 and 2, and the line from B to A is also the bisector of the angle between 2 and 3. The lines from C and B intersect at the point P, and the lines from A and B intersect at the point Q. From P and Q, you can see that the ratios of the distances

## Homework Statement

In triangle ABC, there is a maximun circle (3 intersection points) such that the lengths of the triangles are 3 : 4 : 5. A ray from the smallest angle C is tangential to the opposite side. Another ray from the greatest angle B is also tangential to the opposite side.

Find p : q where p is the length between the ray from the greatest angle to the tangential point and the intersection of this ray to another ray from smaller angle;
q is the length between the intersection of two rays and the vertex C.

## The Attempt at a Solution

The problem can be solved by Mass point geometry and areas/lengths.
However, I am not sure about the ratios of how the rays divide triangle's two sides.

We do know that whether the rays are bisectors although they meet inside the circle.

How would you solve the problem?

I don't follow your description of the problem. Is there a picture you can upload? I don't know where the extra triangles are, unless that was a typo in the first line.

Hogger said:
I don't follow your description of the problem. Is there a picture you can upload? I don't know where the extra triangles are, unless that was a typo in the first line.

The problem in the original form:

A circle inscribed in a 3-4-5 triangle. A segment is drawn from the smaller acute angle to the point of tangency on the opposite side. This segment is divided in the ratio p : q by the segment drawn from the larger acute angle to the point of tangency on its opposite side. If p > q then find p : q.

The same problem is in the following picture at
http://dl.getdropbox.com/u/175564/problem2.JPG [Broken]
You can see a triangle which shows you how I understand the question.

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## 1. What are incircle tangents in a triangle?

Incircle tangents refer to the lines that are tangent to the incircle of a triangle, meaning they touch the incircle at exactly one point. These lines are formed by extending the sides of the triangle until they intersect with the incircle.

## 2. How do you find the length of the incircle tangents in a triangle?

The length of the incircle tangents can be calculated using the formula: t = √(s(s-a)(s-b)(s-c))/s, where s is the semi-perimeter of the triangle (s = (a+b+c)/2) and a, b, and c are the lengths of the sides of the triangle.

## 3. What is the relationship between the incircle tangents and the sides of the triangle?

The incircle tangents are equal in length to each other and are also equal to the distance from the incenter (center of the incircle) to any of the sides of the triangle.

## 4. Can you have a triangle with no incircle tangents?

Yes, a triangle with all sides of equal length (equilateral triangle) does not have any incircle tangents, as the incircle touches all three sides at the same point.

## 5. How are incircle tangents used in geometry?

Incircle tangents have various applications in geometry, including finding the incenter of a triangle, determining the radius of the incircle, and solving problems related to inscribed and circumscribed circles in a triangle.