Lovely Mae M. Piscador

Problem number 8

Problem: Given the center of the inscribed circle and two of the escribed circles, construct a triangle.

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Problem

Given the center of the inscribed circle, and two of the escribed circles, construct a triangle.

Related Concepts

• Inscribed circle - In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches the three sides. The center of the incircle is called the triangle's incenter. The radius of the incircle is called the inradius.

• Escribed circle - An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides. The center of the excircle is called the excenter. The radius of the excircle is called the exradius.

Steps in Drawing Incircle and Excircle of a Triangle

*Incircle of a Triangle(Inscribed)

1. We start with the given triangle ABC.
2. Bisect two angles ( Suppose Angle B and Angle C).
3. Their intersection (D) would be the incenter of the triangle.
4. From the incenter, draw a line perpendicular to segment BC.
5. The segment joining the incenter of the triangle and the intersection of the perpendicular line and segment BC will be the inradius.
6. Draw a circle with the radius DE.

*Excircle of a Triangle(Escribed)

1. We start with the given triangle ABC.
2. Form an exterior angle by extending the adjacent side.
3. Bisect the two exterior angles.
4. The intersection of the two bisector would be the excenter (F).
5. From the excenter, draw a line perpendicular to one of the exterior sides.
6. The segment joining the excenter of the triangle and the intersection (G) of the perpendicular line and exterior side will be the exradius.
7. Draw a circle with the radius of FG.

Related Topics

• Orthocentric system - The center of the incircle can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle and the external bisectors of the other two. Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.

Procedure in solving using Geogebra Software

Step no.1 Open Geogebra application, click graphics then show/hide the axes,

now it will look like this:

Step no.2 Use the slider icon to set the angles of the triangle (so that you’ll be able to move the angles of a triangles in whatever angle you want), then click angle with the given size to apply your slider. You can now draw your triangle using the polygon icon.

Step no.3 Construct the inscribed and two escribed circles of a triangle (For the construction, check the * Steps in Drawing Incircle and Excircle of a Triangle).

Step no. 4 Hide all unnecessary lines, points, and circles in the construction. In the Algebra window, click the button beside the unneeded conic, points, lines, and segments to hide it.

Step no.5 You can now construct the triangle by drawing a segment connecting the incenter (the center of the inscribed circle) and the two excenter (the center of the two escribed circles).