FEUERBACH S THEOREM ON RIGHT TRIANGLE WITH AN EXTENSION

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1 TRTOL JOURL O GOTRY Vol. 6 (2017), o. 2, URH S THOR O RGHT TRGL WTH XTSO TR UG HUG bstract. Given a triangle, we shall construct a circle which is tangent to both the incircle and the circumcircle at a vertex of the triangle. This construction can be considered as the generalization of euerbach s Theorem on right triangle. 1. ntroduction euerbach s Theorem is known as one of the most important theorems with many applications in elementary geometry, see [1,2,3,4,5,6] Theorem 1.1 (euerbach, 1822). n a nonequilateral triangle, the ninepoint circle is internally tangent to the incircle. Recently, ichael J. G. Scheer has given a simple vector proof of this theorem [7]. or historical details of this theorem, please see [3]. n this article, we study a special case of euerbach s Theorem when our triangle is right. eywords and phrases: Right Triangle, euerbach, idline (2010)athematics Subject lassification: 5104, 5125 Received: n revised form: ccepted:

2 104 Tran uang Hung f triangle is right at a vertex, this vertex also is the feets of the altitudes from the rest vertexs, therefore the nine-point circle of this triangle passes through that vertex. We have the specical case of euerbach s Theorem on right triangle as following Theorem 1.2 (euerbach s Theorem on right triangle). Let be a right triangle at., respectively are midpoints of the segments,. Then circumcircle of the triangles is tangent to the incircle of triangle. n this article, we shall present a construction of the circle that is tangent to both the incircle and the circumcircle of triangle. This can be considered as the generalization of theorem 2. irst of all, we shall point out another way to construct the midline in a right triangle by the following theorem Theorem 1.3. Let be a right triangle inscribed circle (O) with diameter. ncirle () of triangle touches, at,, reps. intersects (O) again at. The circle () passes through, which intersects (O) at two points,. Then the line contains the -midline of triangle.

3 euerbach s theorem on right triangle with an extension 105 ow using this construction on right triangle. We establish the following interesting reuslt. Theorem 1.4. Let be a triangle inscribed circle (O). ncirle () of triangle touches, at,, reps. intersects (O) again at. The circle () passes through, which intersects (O) at two points,. The line interects sides, at,, resp. Then the circumcircle of triangle is tangent to () and (O). O Remark 1.1. n the theorem 1.4, if triangle is right at and we apply the construction of midline in the theorem 1.3, we get the theorem 1.2. So we see that the theorem 1.2 as a special case of the theorem The proof of the main theorem We fist present the proof of the theorem 1.4. roof. Let J be the -excenter of the triangle. oint H is the feet of altitude from on line. The segment L is diameter of circle (O). L intersects, at S,T, respectively. Denote by R the midpoint of. Using ythagorean theorem and the metric relations on right triangle, we have the following indenitites ST.L = T L S L = 2 2 = 2 2 = R 2 +R 2 2

4 106 Tran uang Hung = (R )(R+)+R R = R (R+J)+R R = R(RJ +R) = R J. This follows that L J = R ST (1). L R O T H S J all by R the radius of the circumcircle (O). ote that, two triangles and J are similar. We have the indentities H L = H 2R = 2[] 2 2R = 2R = = J. 4R Which implies L J = H (2). rom (1) and (2), we deduce that R ST = H. Hence,

5 euerbach s theorem on right triangle with an extension 107 R = ST H = =. Therefore, we have the equal ratios R = = = k. onsider the homothety center with the ratio k. This homothety transforms the circumcircle of triangle to the circumcircle (O) of triangle, the incircle () transforms to -mixtilinear incircle of. ecause of the touching of -mixtilinear incircle the circumcircle (O), the circumcircle of triangle and the incircle () of are tangent. retty obvious that is parallel to so the circumcircle of triangle is tangent to (O) at. We complete the proof. Remark 2.1. The proof of the theorem 1.4 also contains the proof of the theorem 3. ndeed, in the theorem 3, let the line intersect the sides, at,, respectively. Let the intersection of and be the point R. ecause triangle is right at, R is midpoint of. Using the last equal ratios in the proof of the theorem 1.4, we have = = R = 1 2. R Thus, are midpoints of,. This means the line contains the -midline of triangle. 3. onclusion n this article, we have reiterated the famous theorem of euerbach. Then we apply the euerbach s theorem on the right triangle. Using a new construction of midline in the right triangle, we extended euerbach s theorem on the right triangle and gave a purely geometric solution.

6 108 Tran uang Hung References [1] oxeter, H.S.., ntroduction to Geometry, 2nd edition, John Wiley & Sons, Hoboken,.J., [2] oxeter, H.S.. and Greitzer, S. L., Geometry Revisited, The ath. ssoc. of merica, [3] Yiu,., ntroduction to the Geometry of the Triangle, lorida tlantic University Lecture otes, 2001; with corrections, 2013, available at [4] acay, J. S., History of the nine point circle, roc. dinb. ath. Soc., 11 (1892) [5] arbu,.., undamental theorems of triangle geometry (Romanian), ditura Unique, acău, [6] ogomolny,., euerbach s Theorem, nteractive athematics iscellany and uzzles, [7] Scheer,. J. G., simple vector proof of euerbachs theorem, orum Geometricorum, 11 (2011) HGH SHOOL OR GTD STUDTS, HO UVRSTY O S, HO TOL UVRSTY, HO, VT. -mail address: analgeomatica@gmail.com