(* ========================================================================= *) (* Feuerbach's theorem. *) (* ========================================================================= *) needs "Multivariate/convex.ml";; (* ------------------------------------------------------------------------- *) (* Algebraic condition for two circles to be tangent to each other. *) (* ------------------------------------------------------------------------- *) let CIRCLES_TANGENT = prove (`!r1 r2 c1 c2. &0 <= r1 /\ &0 <= r2 /\ (dist(c1,c2) = r1 + r2 \/ dist(c1,c2) = abs(r1 - r2)) ==> c1 = c2 /\ r1 = r2 \/ ?!x:real^2. dist(c1,x) = r1 /\ dist(c2,x) = r2`, MATCH_MP_TAC REAL_WLOG_LE THEN CONJ_TAC THENL [REPEAT GEN_TAC THEN MATCH_MP_TAC(MESON[] `(!x y. P x y <=> Q y x) ==> ((!x y. P x y) <=> (!x y. Q x y))`) THEN MESON_TAC[DIST_SYM; REAL_ADD_SYM; REAL_ABS_SUB]; ALL_TAC] THEN REPEAT GEN_TAC THEN DISCH_TAC THEN REPEAT GEN_TAC THEN ASM_CASES_TAC `r1 = &0` THENL [ASM_REWRITE_TAC[DIST_EQ_0; MESON[] `(?!x. a = x /\ P x) <=> P a`] THEN REWRITE_TAC[DIST_SYM] THEN REAL_ARITH_TAC; ALL_TAC] THEN ASM_CASES_TAC `r2 = &0` THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN ASM_SIMP_TAC[REAL_ARITH `r1 <= r2 ==> abs(r1 - r2) = r2 - r1`] THEN ASM_REWRITE_TAC[REAL_LE_LT] THEN STRIP_TAC THENL [DISJ2_TAC THEN REWRITE_TAC[EXISTS_UNIQUE] THEN EXISTS_TAC `c1 + r1 / (r1 + r2) % (c2 - c1):real^2` THEN CONJ_TAC THENL [REWRITE_TAC[dist; VECTOR_ARITH `c1 - (c1 + a % (x - y)):real^2 = a % (y - x)`; VECTOR_ARITH `z - (x + a % (z - x)):real^N = (a - &1) % (x - z)`] THEN ASM_REWRITE_TAC[NORM_MUL; GSYM dist] THEN ASM_SIMP_TAC[REAL_ABS_DIV; REAL_ABS_NEG; REAL_FIELD `&0 < r1 /\ &0 < r2 ==> r1 / (r1 + r2) - &1 = --r2 / (r1 + r2)`] THEN ASM_SIMP_TAC[real_abs; REAL_LT_IMP_LE; REAL_LT_ADD] THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD; X_GEN_TAC `y:real^2` THEN STRIP_TAC THEN SUBGOAL_THEN `(y:real^2) IN segment[c1,c2]` MP_TAC THENL [ASM_REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between] THEN ASM_MESON_TAC[DIST_SYM]; REWRITE_TAC[IN_SEGMENT]] THEN DISCH_THEN(X_CHOOSE_THEN `u:real` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN DISCH_THEN SUBST_ALL_TAC THEN UNDISCH_TAC `dist(c1:real^2,(&1 - u) % c1 + u % c2) = r1` THEN REWRITE_TAC[VECTOR_ARITH `(&1 - u) % c1 + u % c2:real^N = c1 + u % (c2 - c1)`] THEN REWRITE_TAC[NORM_ARITH `dist(x:real^2,x + y) = norm y`] THEN ONCE_REWRITE_TAC[GSYM NORM_NEG] THEN REWRITE_TAC[VECTOR_ARITH `--(a % (x - y)):real^N = a % (y - x)`] THEN ASM_REWRITE_TAC[NORM_MUL; GSYM dist; real_abs] THEN DISCH_TAC THEN AP_TERM_TAC THEN AP_THM_TAC THEN AP_TERM_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD]; ASM_CASES_TAC `r1:real = r2` THENL [ASM_MESON_TAC[REAL_SUB_REFL; DIST_EQ_0]; DISJ2_TAC] THEN SUBGOAL_THEN `r1 < r2` ASSUME_TAC THENL [ASM_REAL_ARITH_TAC; ALL_TAC] THEN REWRITE_TAC[EXISTS_UNIQUE] THEN EXISTS_TAC `c2 + r2 / (r2 - r1) % (c1 - c2):real^2` THEN CONJ_TAC THENL [REWRITE_TAC[dist; VECTOR_ARITH `c1 - (c1 + a % (x - y)):real^2 = --(a % (x - y)) /\ c1 - (c2 + a % (c1 - c2)):real^2 = (&1 - a) % (c1 - c2)`] THEN ASM_REWRITE_TAC[NORM_MUL; NORM_NEG; GSYM dist] THEN ASM_SIMP_TAC[REAL_ABS_DIV; REAL_ABS_NEG; REAL_FIELD `r1 < r2 ==> &1 - r2 / (r2 - r1) = --(r1 / (r2 - r1))`] THEN ASM_SIMP_TAC[real_abs; REAL_SUB_LE; REAL_LT_IMP_LE] THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD; X_GEN_TAC `y:real^2` THEN STRIP_TAC THEN SUBGOAL_THEN `(c1:real^2) IN segment[c2,y]` MP_TAC THENL [ASM_REWRITE_TAC[GSYM BETWEEN_IN_SEGMENT; between] THEN ONCE_REWRITE_TAC[DIST_SYM] THEN ASM_REAL_ARITH_TAC; REWRITE_TAC[IN_SEGMENT]] THEN DISCH_THEN(X_CHOOSE_THEN `u:real` MP_TAC) THEN REPEAT(DISCH_THEN(CONJUNCTS_THEN2 ASSUME_TAC MP_TAC)) THEN ASM_CASES_TAC `u = &0` THENL [ASM_REWRITE_TAC[VECTOR_MUL_LZERO; VECTOR_ADD_RID; REAL_SUB_RZERO] THEN REWRITE_TAC[VECTOR_MUL_LID] THEN ASM_MESON_TAC[DIST_EQ_0; REAL_SUB_0]; ALL_TAC] THEN DISCH_THEN SUBST_ALL_TAC THEN UNDISCH_TAC `dist((&1 - u) % c2 + u % y:real^2,c2) = r2 - r1` THEN REWRITE_TAC[VECTOR_ARITH `(&1 - u) % c1 + u % c2:real^N = c1 + u % (c2 - c1)`] THEN REWRITE_TAC[NORM_ARITH `dist(x + y:real^2,x) = norm y`] THEN ONCE_REWRITE_TAC[GSYM NORM_NEG] THEN REWRITE_TAC[VECTOR_ARITH `--(a % (x - y)):real^N = a % (y - x)`] THEN ASM_REWRITE_TAC[NORM_MUL; GSYM dist; real_abs] THEN REWRITE_TAC[VECTOR_ARITH `c + v % ((c + u % (y - c)) - c):real^2 = c + v % u % (y - c)`] THEN DISCH_THEN(SUBST1_TAC o SYM) THEN REWRITE_TAC[VECTOR_MUL_EQ_0; VECTOR_ARITH `y:real^2 = c + u % v % (y - c) <=> (&1 - u * v) % (y - c) = vec 0`] THEN DISJ1_TAC THEN REPEAT(POP_ASSUM MP_TAC) THEN CONV_TAC REAL_FIELD]]);; (* ------------------------------------------------------------------------- *) (* Feuerbach's theorem *) (* *) (* Given a non-degenerate triangle abc, let the circle passing through *) (* the midpoints of its sides (the "9 point circle") have center "ncenter" *) (* and radius "nradius". Now suppose the circle with center "icenter" and *) (* radius "iradius" is tangent to three sides (either internally or *) (* externally to one side and two produced sides), meaning that it's either *) (* the inscribed circle or one of the three escribed circles. Then the two *) (* circles are tangent to each other, i.e. either they are identical or they *) (* touch at exactly one point. *) (* ------------------------------------------------------------------------- *) let FEUERBACH = prove (`!a b c mbc mac mab pbc pac pab ncenter icenter nradius iradius. ~(collinear {a,b,c}) /\ midpoint(a,b) = mab /\ midpoint(b,c) = mbc /\ midpoint(c,a) = mac /\ dist(ncenter,mbc) = nradius /\ dist(ncenter,mac) = nradius /\ dist(ncenter,mab) = nradius /\ dist(icenter,pbc) = iradius /\ dist(icenter,pac) = iradius /\ dist(icenter,pab) = iradius /\ collinear {a,b,pab} /\ orthogonal (a - b) (icenter - pab) /\ collinear {b,c,pbc} /\ orthogonal (b - c) (icenter - pbc) /\ collinear {a,c,pac} /\ orthogonal (a - c) (icenter - pac) ==> ncenter = icenter /\ nradius = iradius \/ ?!x:real^2. dist(ncenter,x) = nradius /\ dist(icenter,x) = iradius`, REPEAT GEN_TAC THEN DISCH_TAC THEN MATCH_MP_TAC CIRCLES_TANGENT THEN POP_ASSUM MP_TAC THEN MAP_EVERY (fun t -> ASM_CASES_TAC t THENL [ALL_TAC; ASM_MESON_TAC[DIST_POS_LE]]) [`&0 <= nradius`; `&0 <= iradius`] THEN ASM_REWRITE_TAC[dist; NORM_EQ_SQUARE] THEN ASM_SIMP_TAC[REAL_LE_ADD; REAL_ABS_POS; GSYM NORM_POW_2; GSYM dist] THEN REWRITE_TAC[REAL_POW2_ABS] THEN POP_ASSUM_LIST(K ALL_TAC) THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d /\ e <=> b /\ c /\ d /\ a /\ e`] THEN GEOM_ORIGIN_TAC `a:real^2` THEN GEOM_NORMALIZE_TAC `b:real^2` THEN CONJ_TAC THENL [REWRITE_TAC[INSERT_AC; COLLINEAR_2]; ALL_TAC] THEN GEOM_BASIS_MULTIPLE_TAC 1 `b:real^2` THEN SIMP_TAC[NORM_MUL; NORM_BASIS; DIMINDEX_2; ARITH; real_abs] THEN GEN_TAC THEN DISCH_THEN(K ALL_TAC) THEN REWRITE_TAC[REAL_MUL_RID] THEN DISCH_THEN SUBST1_TAC THEN REWRITE_TAC[VECTOR_MUL_LID] THEN REPEAT GEN_TAC THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o SYM) MP_TAC)) THEN REWRITE_TAC[COLLINEAR_3_2D] THEN REWRITE_TAC[orthogonal; dist; NORM_POW_2] THEN ASM_REWRITE_TAC[midpoint] THEN REWRITE_TAC[DOT_2; DOT_LSUB; DOT_RSUB] THEN SIMP_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; VEC_COMPONENT; VECTOR_MUL_COMPONENT; BASIS_COMPONENT; DIMINDEX_2; ARITH] THEN CONV_TAC REAL_RING);; (* ------------------------------------------------------------------------- *) (* As a little bonus, verify that the circle passing through the *) (* midpoints of the sides is indeed a 9-point circle, i.e. it passes *) (* through the feet of the altitudes and the midpoints of the lines joining *) (* the vertices to the orthocenter (where the alititudes intersect). *) (* ------------------------------------------------------------------------- *) let NINE_POINT_CIRCLE_1 = prove (`!a b c:real^2 mbc mac mab fbc fac fab ncenter nradius. ~(collinear {a,b,c}) /\ midpoint(a,b) = mab /\ midpoint(b,c) = mbc /\ midpoint(c,a) = mac /\ dist(ncenter,mbc) = nradius /\ dist(ncenter,mac) = nradius /\ dist(ncenter,mab) = nradius /\ collinear {a,b,fab} /\ orthogonal (a - b) (c - fab) /\ collinear {b,c,fbc} /\ orthogonal (b - c) (a - fbc) /\ collinear {c,a,fac} /\ orthogonal (c - a) (b - fac) ==> dist(ncenter,fab) = nradius /\ dist(ncenter,fbc) = nradius /\ dist(ncenter,fac) = nradius`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d /\ e <=> b /\ c /\ d /\ a /\ e`] THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o SYM) MP_TAC)) THEN ASM_REWRITE_TAC[dist; NORM_EQ_SQUARE; REAL_POS] THEN REWRITE_TAC[COLLINEAR_3_2D] THEN REWRITE_TAC[orthogonal; dist; NORM_POW_2] THEN ASM_REWRITE_TAC[midpoint] THEN REWRITE_TAC[DOT_2; DOT_LSUB; DOT_RSUB] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT] THEN SIMP_TAC[] THEN CONV_TAC REAL_RING);; let NINE_POINT_CIRCLE_2 = prove (`!a b c:real^2 mbc mac mab fbc fac fab ncenter nradius. ~(collinear {a,b,c}) /\ midpoint(a,b) = mab /\ midpoint(b,c) = mbc /\ midpoint(c,a) = mac /\ dist(ncenter,mbc) = nradius /\ dist(ncenter,mac) = nradius /\ dist(ncenter,mab) = nradius /\ collinear {a,b,fab} /\ orthogonal (a - b) (c - fab) /\ collinear {b,c,fbc} /\ orthogonal (b - c) (a - fbc) /\ collinear {c,a,fac} /\ orthogonal (c - a) (b - fac) /\ collinear {oc,a,fbc} /\ collinear {oc,b,fac} /\ collinear{oc,c,fab} ==> dist(ncenter,midpoint(oc,a)) = nradius /\ dist(ncenter,midpoint(oc,b)) = nradius /\ dist(ncenter,midpoint(oc,c)) = nradius`, REPEAT GEN_TAC THEN ONCE_REWRITE_TAC[TAUT `a /\ b /\ c /\ d /\ e <=> b /\ c /\ d /\ a /\ e`] THEN REPLICATE_TAC 3 (DISCH_THEN(CONJUNCTS_THEN2 (ASSUME_TAC o SYM) MP_TAC)) THEN ASM_REWRITE_TAC[dist; NORM_EQ_SQUARE; REAL_POS] THEN REWRITE_TAC[COLLINEAR_3_2D] THEN REWRITE_TAC[orthogonal; dist; NORM_POW_2] THEN ASM_REWRITE_TAC[midpoint] THEN REWRITE_TAC[DOT_2; DOT_LSUB; DOT_RSUB] THEN REWRITE_TAC[VECTOR_ADD_COMPONENT; VECTOR_SUB_COMPONENT; VECTOR_MUL_COMPONENT; VEC_COMPONENT] THEN SIMP_TAC[] THEN CONV_TAC REAL_RING);;