Question 252997: prove that common internal tangent segments of two circles are congruent.
Answer by drk(1908)   (Show Source):
You can put this solution on YOUR website! This might be a bit wild to explain via description, but here we go . . .
First, draw two circles and call the left one center (A) and the right one center (B).
Second, draw to internal tangents. One tangent touches (B) at point x and it touches (A) at point y. The second tangent touches (B) at point q and touches (A) at point p.
Third, where the two tangents cross, call that point e.
We are now ready to prove.
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STATEMENT 1: two circles, A and B that are internally tangent at point e, and touch a at Y and p as well as touch B at x and q.
REASON 1: given
STATEMENT 2: Ay congruent to Ap; Bq congruent to Bx.
REASON 2: definition of radii
STATEMENT 3: angle Aye, angle Ape, angle Bqe, and angle Bxe are all right angles
REASON 3: definition of tangent line
STATEMENT 4: Ae congruent Ae ; Be congruent Be
REASON 4: reflexive property of equality
STATEMENT 5: triangle Ape congruent triangle Aye ; triangle Bqe congruent triangle Bxe
REASON 5: HL postulate
STATEMENT 6: ey congruent ep ; eq congruent ex
REASON 6: CPCTC
STATEMENT 7: ye + ex = yx ; pe + eq = pq
REASON 7: segment addition postulate
STATEMENT 8: ye + ex = pq ; pe + eq = xy
REASON 8: substitution [6,7]
STATEMENT 9: pq = xy
REASON 9: substitution [7,8]
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