G.104 touching circles have distinct centers

III.6

Two circles that touch one another do not have the same center.

touching_circles_distinct_centers(( A B C ), ( C D E ), C)

given ( A B C ), ( C D E ) touching at C

prove center(( A B C )) ≠ center(( C D E ))

∵ circles_said_touch_defgiven ( A B C ), ( C D E ) touching at C

∵ segmentconstruct assume common center F; segment(F, C) → / F C /, draw / F E B /

∵ circle_radii, circle_centerassert F = center(( A B C )) → / F C / = / F B /

∵ circle_radii, equal_transitive, whole_greater_than_partassert F = center(( C D E )) → / F C / = / F E / → / F E / = / F B / (< > ⇒⇐)

∴ center(( A B C )) ≠ center(( C D E ))

Dependency Graph