1. Construct a Triangle Given the Length of the Base, the Difference of the Base Angles and the Foot of the Altitude to the Base

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This Demonstration constructs a triangle given the length of its base, the difference of the base angles and the foot of the altitude from to the base.

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Construction

Step 1: Draw a horizontal line of length and the foot of the altitude from . Measure out a point so that , so that is the midpoint of .

Step 2: Draw a vertical line through .

Step 3: Draw a circle with center so that the central angle .

Step 4: Let the intersections of and , if they exist, be and . Join , , and .

Then either of the two triangles or is a solution.

Step 5: Draw the triangle (dashed line).

Verification

Let and .

Consider the triangle (the verification for the triangle is similar). Because is the right bisector of , , so the angle . But this angle subtends the chord and is half of the central angle over , which is . So .

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Contributed by: Izidor Hafner (August 2017)
Open content licensed under CC BY-NC-SA


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