How to Determine the Incircle of a Triangle Given Two Sides and an Angle

Suppose you are given the lengths of two sides and one angle (not the angle opposite the given side) in a triangle, but you do not know the third side. In this case, you can determine the incircle of the triangle, the circle that touches all three sides of the triangle. Here’s how to do it step by step.

Step 1: Determine the Angle Bisectors

The incenter of a triangle is the point where the angle bisectors of the triangle intersect. The incenter is equidistant from all three sides of the triangle. This means that for the given angle, the incenter must lie on the angle bisector of that angle.

To find the angle bisectors, start by drawing arcs with equal radii on the adjacent sides to the given angle. Then, draw more arcs from these points with the same radius. The intersections of these arcs will give you the points where the angle bisectors cross the sides.

Draw two arcs of equal radius from the endpoints of the given side on the adjacent sides. Mark the intersection of these arcs. This point lies on the angle bisector. Repeat the process for the other two angles in the triangle.

With the angle bisectors determined, you can now find the incenter by finding the intersection of these bisectors.

Step 2: Determine the Third Side (If Not Already Known)

Since you know the lengths of two sides and one angle, you can use this information to find the third side using the Law of Cosines. The Law of Cosines states:

cos(A) (b2 c2 - a2) / (2bc)

where a, b, and c are the lengths of the sides of the triangle, and A is the angle opposite side a. Rearrange this formula to solve for the unknown side:

a sqrt(b2 c2 - 2bc cos(A))

Once you have the third side, you can proceed to the next step.

Step 3: Find the Coordinates of the Vertices

For simplicity, you can set one of the vertices of the triangle at the origin (0,0) and the angle at the origin. Let’s denote the sides as follows:

Side KL with length a, lying on the x-axis (x a). Side KM with length b, making an angle A with the x-axis. Side LM as the unknown side with length c.

The coordinates of the vertices L and M are:

L’s coordinates: (a, 0) M’s coordinates: (b cos(A), b sin(A))

Step 4: Calculate the Vector LM

The vector LM can be calculated using the coordinates of L and M:

LM (b cos(A) - a, b sin(A))

Let p b cos(A) - a and q b sin(A).

Step 5: Determine the Angles A' and B'

The angle A' is found using the tangent function:

A' arctan(p / q)

If the result of the arctan function is a negative angle, add 180 degrees to get the actual value. Similarly, angle B' is:

B' 2 - A' (if A' is negative, subtract the negative)

Step 6: Find the Equations of the Angle Bisectors

The equation of the angle bisector of angle A is:

y (sin(A'/2) / cos(A'/2)) x

This can be simplified to:

y x tan(A'/2)

The equation of the angle bisector of angle B is:

y -tan(B'/2) x d

where d is the length of side KL (a).

Step 7: Find the Incenter (Intersection of Bisectors)

The incenter is the point where the two angle bisectors intersect. To find the incenter, solve the system of equations:

-tan(B'/2) x a x tan(A'/2)

Solve for x to get the x-coordinate of the incenter, and substitute x back into one of the equations to get the y-coordinate.

Step 8: Determine the Radius of the Incircle

The radius of the incircle (r) can be found using the coordinates of the incenter (v, w) and the formula:

(g - v)2 (h - w)2 w2

Simplify to:

(g - v)2 - 2gv (h - w)2 0

The incenter (v, w) is the center of the incircle, and the radius (r) is w.

Conclusion

By following these steps, you can determine the incircle of a triangle given two sides and an angle. While the steps may seem complex, they provide a systematic approach to finding the incenter and the radius of the incircle. This method is particularly useful for solving geometric problems and can be applied in various real-world scenarios, such as in architecture, engineering, and design.