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To inscribe a circle in a given triangle |
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To inscribe a circle in a given triangle |
Given:  ABC |
| Required: To inscribe a circle
in ABC |
| Procedure: |
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Construct ray AD bisecting  A.
(help) or use Sketchpad commands: |
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Select vertices C, A, B in that order. Use the angle bisector
command under the construct menu and construct the bisector. |
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Label the bisector ray AD. |
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Repeat either of the above constructions to construct ray BE, the
bisector of B. |
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Label the point of intersection of the bisectors
point O. Bisectors of the 's
of a
are concurrent in a point equidistant from the side of the triangle. |
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Drop a perpendicular line from O to AB. (help) or use Sketchpad
commands: |
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Select point O and segment AB, use the perpendicular command under the
construct menu and construct a perpendicular at O. |
Label the point of intersection of the  line
and segment AB, point F. |
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Use the segment command under the construct menu and construct segment
OF. |
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Use the circle by center+ radius command under the construct menu and
construct a circle with O as center and radius congruent to segment OF. |
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This is the required circle. |
| Proof: |
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AB is tangent to circle O. Similarly, BC and AC are tangent to
circle O. (A line
to the radius at is extremity on the circle is tangent to the circle) |
 circle
O is inscribed in ABC.
(The sides of the
are tangent to the circle) |
| Sketch: In the sketch below drag the vertices of the triangle around. Do you see how the circle remains inside changing in size as necessary. What condition is not being met when one of the angle bisectors and thus the circle disappears? |
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This page uses JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright © 1990-2001 by KCP Technologies, Inc. Licensed only for non-commercial use. inscribe a circle in a triangle |
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