Two Triads of Congruent Circles from Reflections

Transcription

1 Forum Geometriorum Volume 8 (2008) FRUM GEM SSN Two Trids of ongruent irles from Refletions Qung Tun ui strt. Given tringle, we onstrut two trids of ongruent irles through the verties, one ssoited with refletions in the ltitudes, nd the other refletions in the ngle isetors. 1. Refletions in the ltitudes Given tringle with orthoenter, let nd e the refletions of nd in the line. These re points on the sideline so tht =. Similrly, onsider the refletions, of, respetively in the line, nd, of, in the line. Theorem 1. The irles,, nd re ongruent. Figure 1. Proof. Let e the irumenter of tringle, nd X its refletion in the -ltitude. This is the irumenter of tringle, the refletion of tringle in its -ltitude. See Figure 2. t follows tht lies on the perpendiulr isetor of X, nd X =. Similrly, if Y nd Z re the refletions of in the lines nd respetively, then Y = Z =. t follows tht, X, Y, Z re onyli, nd is the enter of the irle ontining them. See Figure 3. Pulition Dte: Jnury 14, ommuniting Editor: Pul Yiu. The uthor thnks Pul Yiu for his help in the preprtion of this pper.

2 8 Q. T. ui Y X X Z Figure 2 Figure 3 Let e the irumenter of tringle. Note the equlities of vetors X = =, Y = =, Z = =. The three tringles,, nd re the trnsltions of Y Z y, ZX y, nd XY y respetively. Y X Z Figure 4. Therefore, the irumirles of the three tringles re ll ongruent nd hve rdius. Their enters re the trnsltions of y the three vetors.

3 Two trids of ongruent irles from refletions 9 2. Refletions in the ngle isetors Let e the inenter of tringle. onsider the refletions of the verties in the ngle isetors:, of, in,, of, in, nd, of, in. See Figure 5. Theorem 2. The irles,, nd re ongruent. Figure 5. Proof. onsider the refletions, of, nd, of, in. See Figure 6. in,, of, in, X Y Z Figure 6 Figure 7

4 10 Q. T. ui Note the equlities of vetors =, =, =. With the irumenter of tringle, these define points X, Y, Z suh tht X = =, Y = =, Z = =. The tringles, nd re the trnsltions of Y Z, Z X nd X Y y the vetors, nd respetively. See Figure 7. Note, in Figure 8, tht X is symmetri trpezoid nd = =. t follows tht tringles X nd re ongruent, nd X =. Similrly, Y = nd Z =. This mens tht the four points, X, Y, Z re on irle enter. See Figure 9. The irumenters,, of the tringles, nd re the trnsltions of y these vetors. These irumirles re ongruent to the irle (). X X Y Z Figure 8 Figure 9 The segments, nd re prllel nd equl in lengths. The tringles, nd re the refletions of, nd in the respetive ngle isetors. See Figure 10. t follows tht their irumirles re ll ongruent to (). Let,, e the irumenters of tringles, nd respetively. The lines nd re symmetri with respet to the isetor of ngle. Sine, nd re prllel to the line, the refletions in the ngle isetors onur t the isogonl onjugte of the infinite point of. This is point P on the irumirle. t is the tringle enter X 104 in [1]. Finlly, sine = =, we lso hve = =. The 6 irumenters ll lie on the irle, enter, rdius R.

5 Two trids of ongruent irles from refletions 11 P Y X Z Figure 10. To onlude this note, we estlish n interesting property of the enters of the irles in Theorem 2. Proposition 3. The lines, nd re perpendiulr to, nd respetively. M Figure 11.

6 12 Q. T. ui Proof. t is enough to prove tht for the line. The other two ses re similr. Let M e the intersetion (other thn ) of the irle ( ) with the irumirle of tringle. Sine = M (irumrdius) nd M =, M is prllelogrm. This mens tht M = =, nd M is lso prllelogrm. From this we onlude tht M, eing prllel to, is perpendiulr to the isetor. Thus, M is the midpoint of the r, nd M is perpendiulr to. Sine = M, the line is lso perpendiulr to. Sine the six irles ( ) nd ( ) et re ongruent (with ommon rdius ) nd their enters re ll t distne R from, it is ler tht there re two irles, enter, tngent to ll these irles. These two irles re tngent to the irumirle, the point of tngeny eing the intersetion of the irumirle with the line. These re the tringle enters X 1381 nd X 1382 of [1]. X 1381 X 1382 Figure 12. Referenes [1]. Kimerling, Enylopedi of Tringle enters, ville t Qung Tun ui: 45, 296/86 y-street, Minh Khi Street, noi, Vietnm E-mil ddress: qtun1962@yhoo.om