G.114 equal chords are equidistant from the center

III.14

In a circle, chords are equal if and only if they are equally distant from the center.
equal_chords_equidistant(( A B D C ), / A B /, / C D /)
given / A B /, / C D / chords in ( A B D C )
prove / A B / = / C D / ⇔ / E F / = / E G /
A B C D E F G
∵ circle_segments_said_defgiven / A B /, / C D / chords in ( A B D C ), first with / A B / = / C D /
∵ find_circle_center, perpendicular_from, segmentconstruct find_circle_center(( A B D C )) → E, / E F / ⊥ / A B /, / E G / ⊥ / C D /, join / A E /, / E C /
∵ diameter_bisects_chord_perpendicularassert diameter_bisects_chord_perpendicular(/ E F /, / A B /) → / A F / = / F B /, / C G / = / G D / → (/ A B / = / C D / ⇔ / A F / = / C G /)
∵ circle_radii, pythagorean_theoremassert / A E / = / E C / → sq(/ A F /) + sq(/ E F /) = sq(/ A E /) = sq(/ E C /) = sq(/ C G /) + sq(/ E G /)
∵ subtract_equals, circle_segments_said_defassert / A F / = / C G / ⇒ sq(/ E F /) = sq(/ E G /) ⇒ / E F / = / E G / (and conversely / E F / = / E G / ⇒ / A F / = / C G / ⇒ / A B / = / C D /)
∴ / A B / = / C D / ⇔ / E F / = / E G /

Dependency Graph

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"G.21" -> "G.15"; "G.35" -> "G.15"; "G.47" -> "G.15"; "G.17" -> "G.16"; "G.20" -> "G.16"; "G.21" -> "G.16"; "G.37" -> "G.16"; "G.39" -> "G.16"; "G.40" -> "G.16"; "G.47" -> "G.16"; "G.56" -> "G.16"; "G.64" -> "G.16"; "G.66" -> "G.16"; "G.70" -> "G.16"; "G.75" -> "G.16"; "G.20" -> "G.17"; "G.31" -> "G.17"; "G.32" -> "G.17"; "G.34" -> "G.17"; "G.20" -> "G.18"; "G.25" -> "G.18"; "G.39" -> "G.18"; "G.41" -> "G.18"; "G.45" -> "G.18"; "G.53" -> "G.18"; "G.58" -> "G.18"; "G.66" -> "G.18"; "G.20" -> "G.19"; "G.25" -> "G.19"; "G.39" -> "G.19"; "G.40" -> "G.19"; "G.64" -> "G.19"; "G.114" -> "G.19"; "G.21" -> "G.20"; "G.25" -> "G.21"; "G.31" -> "G.21"; "G.34" -> "G.21"; "G.41" -> "G.21"; "G.43" -> "G.21"; "G.45" -> "G.21"; "G.47" -> "G.21"; "G.51" -> "G.21"; "G.75" -> "G.21"; "G.24" -> "G.22"; "G.25" -> "G.22"; "G.28" -> "G.22"; "G.29" -> "G.22"; "G.30" -> "G.22"; "G.31" -> "G.22"; "G.32" -> "G.22"; "G.33" -> "G.22"; "G.34" -> "G.22"; "G.35" -> "G.22"; "G.37" -> "G.22"; "G.39" -> "G.22"; "G.40" -> "G.22"; "G.41" -> "G.22"; "G.43" -> "G.22"; "G.44" -> "G.22"; "G.45" -> "G.22"; "G.48" -> "G.22"; "G.51" -> "G.22"; "G.53" -> "G.22"; "G.55" -> "G.22"; "G.56" -> "G.22"; "G.58" -> "G.22"; "G.62" -> "G.22"; "G.63" -> "G.22"; "G.64" -> "G.22"; "G.75" -> "G.22"; "G.76" -> "G.22"; "G.77" -> "G.22"; "G.78" -> "G.22"; "G.79" -> "G.22"; "G.100" -> "G.22"; "G.24" -> "G.23"; "G.29" -> "G.23"; "G.25" -> "G.24"; "G.32" -> "G.24"; "G.41" -> "G.24"; "G.51" -> "G.24"; "G.62" -> "G.24"; "G.64" -> "G.24"; "G.79" -> "G.24"; "G.28" -> "G.25"; "G.43" -> "G.25"; "G.44" -> "G.25"; "G.45" -> "G.25"; "G.100" -> "G.25"; "G.28" -> "G.26"; "G.39" -> "G.26"; "G.41" -> "G.26"; "G.43" -> "G.26"; "G.45" -> "G.26"; "G.51" -> "G.26"; "G.98" -> "G.26"; "G.29" -> "G.28"; "G.31" -> "G.29"; "G.34" -> "G.29"; "G.35" -> "G.29"; "G.48" -> "G.29"; "G.98" -> "G.29"; "G.100" -> "G.29"; "G.31" -> "G.30"; "G.48" -> "G.30"; "G.32" -> "G.31"; "G.35" -> "G.32"; "G.41" -> "G.32"; "G.98" -> "G.32"; "G.34" -> "G.33"; "G.35" -> "G.33"; "G.37" -> "G.33"; "G.38" -> "G.33"; "G.39" -> "G.33"; "G.40" -> "G.33"; "G.55" -> "G.33"; "G.56" -> "G.33"; "G.75" -> "G.33"; "G.76" -> "G.33"; "G.77" -> "G.33"; "G.78" -> "G.33"; "G.79" -> "G.33"; "G.98" -> "G.33"; "G.100" -> "G.33"; "G.113" -> "G.33"; "G.37" -> "G.34"; "G.75" -> "G.34"; "G.98" -> "G.34"; "G.114" -> "G.35"; "G.37" -> "G.36"; "G.56" -> "G.36"; "G.62" -> "G.36"; "G.64" -> "G.36"; "G.79" -> "G.36"; "G.39" -> "G.37"; "G.40" -> "G.37"; "G.56" -> "G.37"; "G.39" -> "G.38"; "G.40" -> "G.38"; "G.79" -> "G.39"; "G.41" -> "G.40"; "G.56" -> "G.40"; "G.43" -> "G.41"; "G.51" -> "G.41"; "G.53" -> "G.41"; "G.44" -> "G.43"; "G.45" -> "G.44"; "G.47" -> "G.45"; "G.48" -> "G.47"; "G.58" -> "G.48"; "G.62" -> "G.51"; "G.100" -> "G.51"; "G.53" -> "G.52"; "G.56" -> "G.52"; "G.58" -> "G.52"; "G.62" -> "G.52"; "G.64" -> "G.52"; "G.66" -> "G.52"; "G.70" -> "G.52"; "G.75" -> "G.52"; "G.79" -> "G.52"; "G.58" -> "G.53"; "G.56" -> "G.55"; "G.62" -> "G.56"; "G.64" -> "G.56"; "G.75" -> "G.56"; "G.66" -> "G.58"; "G.75" -> "G.58"; "G.79" -> "G.58"; "G.62" -> "G.61"; "G.66" -> "G.61"; "G.70" -> "G.61"; "G.64" -> "G.62"; "G.66" -> "G.62"; "G.70" -> "G.62"; "G.75" -> "G.62"; "G.64" -> "G.63"; "G.75" -> "G.63"; "G.79" -> "G.63"; "G.66" -> "G.64"; "G.70" -> "G.66"; "G.79" -> "G.70"; "G.79" -> "G.75"; "G.78" -> "G.76"; "G.78" -> "G.77"; "G.79" -> "G.78"; "G.114" -> "G.79"; "G.114" -> "G.98"; "G.114" -> "G.100"; "G.114" -> "G.113"; }

Required for

G.116