Chapter 7. Some triangle centers. 7.1 The Euler line Inferior and superior triangles

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1 hapter 7 Some triangle centers 7.1 The Euler line nferior and superior triangles G F E G D The inferior triangle of is the triangle DEF whose vertices are the midpoints of the sides,,. The two triangles share the same centroid G, and are homothetic at G with ratio 1 : 2. The superior triangle of is the triangle bounded by the parallels of the sides through the opposite vertices. The two triangles also share the same centroid G, and are homothetic at G with ratio 2 : 1.

2 216 Some triangle centers The orthocenter and the Euler line The three altitudes of a triangle are concurrent. This is because the line containing an altitude of triangle is the perpendicular bisector of a side of its superior triangle. The three lines therefore intersect at the circumcenter of the superior triangle. This is the orthocenter of the given triangle. H G O The circumcenter, centroid, and orthocenter of a triangle are collinear. This is because the orthocenter, being the circumcenter of the superior triangle, is the image of the circumcenter under the homothety h(g, 2). The line containing them is called the Euler line of the reference triangle (provided it is non-equilateral). The orthocenter of an acute (obtuse) triangle lies in the interior (exterior) of the triangle. The orthocenter of a right triangle is the right angle vertex.

3 7.2 The nine-point circle The nine-point circle Theorem 7.1. The following nine points associated with a triangle are on a circle whose center is the midpoint between the circumcenter and the orthocenter: (i) the midpoints of the three sides, (ii) the pedals (orthogonal projections) of the three vertices on their opposite sides, (iii) the midpoints between the orthocenter and the three vertices. D F E G N O H E F D Proof. (1) Let N be the circumcenter of the inferior triangle DEF. Since DEF and are homothetic at G in the ratio 1 : 2, N, G, O are collinear, and NG : GO = 1 : 2. Since HG : GO = 2 : 1, the four are collinear, and HN : NG : GO = 3 : 1 : 2, and N is the midpoint of OH. (2) Let be the pedal of H on. Since N is the midpoint of OH, the pedal of N is the midpoint of D. Therefore, N lies on the perpendicular bisector of D, and N = ND. Similarly, NE = N, and NF = N for the pedals of H on and respectively. This means that the circumcircle of DEF also contains,,. (3) Let D, E, F be the midpoints of H, H, H respectively. The triangle D E F is homothetic to at H in the ratio 1 : 2. Denote by N its circumcenter. The points N, G, O are collinear, and N G : GO = 1 : 2. t follows that N = N, and the circumcircle of DEF also contains D, E, F. This circle is called the nine-point circle of triangle. ts center N is called the nine-point center. ts radius is half of the circumradius of.

4 218 Some triangle centers Exercise 1. Let H be the orthocenter of triangle. Show that the Euler lines of triangles, H, H and H are concurrent For what triangles is the Euler line parallel (respectively perpendicular) to an angle bisector? 2 3. Prove that the nine-point circle of a triangle trisects a median if and only if the side lengths are proportional to its medians lengths in some order. P F E H N G D Proof. ( ) 1 3 m2 a = 1 4 (b2 + c 2 a 2 ); 4m 2 a = 3(b2 + c 2 a 2 ); 2b 2 + 2c 2 a 2 = 3(b 2 + c 2 a 2 ), 2a 2 = b 2 + c 2. Therefore, is a root-mean-square triangle. 4. Let P be a point on the circumcircle. What is the locus of the midpoint of HP? Why? 5. f the midpoints of P, P, P are all on the nine-point circle, must P be the orthocenter of triangle? 3 1 Hint: find a point common to them all. 2 The Euler line is parallel (respectively perpendicular) to the bisector of angle if and only if α = 120 (respectively 60 ). 3 P. iu and J. oung, Problem 2437 and solution, rux Math. 25 (1999) 173; 26 (2000) 192.

5 7.3 The incircle and the Gergonne point The incircle and the Gergonne point f the incircle of triangle touches,, at,, respectively, = s b s c s c s a s a s b = 1. y eva s theorem, the lines,, are concurrent. The intersection is called the Gergonne point G e of the triangle. s a s a s b G e s c s b s c Triangle is called the intouch triangle of. learly, = + 2 t is always acute angled, and, = + 2, = +. 2 = 2r cos 2, = 2r cos 2, = 2r cos 2.

6 220 Some triangle centers Exercise 1. Given three points,, not on the same line, construct three circles, with centers at,,, mutually tangent to each other externally. 2. Show that if the incenter of triangle lies on the Gergonne cevian, then is isosceles with =. Deduce that the triangle is equilateral if and only if its Gergonne point and incenter coincide. 3. The incircle of triangle touches the sides and at and respectively. Suppose =. Show that the triangle is isosceles.

7 7.3 The incircle and the Gergonne point is a triangle with = 25, = 21, and = 28. The incircle touches at and at. Let D and E be the midpoints of and respectively. Show that (i) the lines and E intersect on the bisector of angle ; (ii) the lines D and also intersect on the bisector of angle. E D P 5. Given triangle, construct a circle tangent to at and at such that the line passes through the centroid G. Show that G : G = c : b. G

8 222 Some triangle centers 6. Suppose the incircle of triangle touches the side at. Show that the incircles of triangles and touch the side at the same point. 7. Two circles are orthogonal to each other if their tangents at an intersection are perpendicular to each other. Given three points,, not on a line, construct three circles with these as centers and orthogonal to each other. (1) onstruct the tangents from to the circle (b), and the circle tangent to these two lines and to (a) internally. (2) onstruct the tangents from to the circle (a), and the circle tangent to these two lines and to (b) internally. (3) The two circles in (1) and (2) are congruent. 8. Given a point on a line segment, construct a right-angled triangle whose incircle touches the hypotenuse at Let be a triangle with incenter. (1a) onstruct a tangent to the incircle at the point diametrically opposite to its point of contact with the side. Let this tangent intersect at 1 and at 1. (1b) Same in part (a), for the side, and let the tangent intersect at 2 and at 2. (1c) Same in part (a), for the side, and let the tangent intersect at 3 and at 3. (2) Note that 3 = 2. onstruct the circle tangent to and at 3 and 2. How does this circle intersect the circumcircle of triangle? 10. The incircle of touches the sides,, at D, E, F respectively. is a point inside such that the incircle of touches at D also, and touches and at and respectively. (1) The four points E, F,, are concyclic. (2) What is the locus of the center of the circle EF? 4 P. iu, G. Leversha, and T. Seimiya, Problem 2415 and solution, rux Math. 25 (1999) 110; 26 (2000)

9 7.4 The excircles and the Nagel point The excircles and the Nagel point Let,, be the points of tangency of the excircles ( a ), ( b ), ( c ) with the corresponding sides of triangle. The lines,, are concurrent. The common point N a is called the Nagel point of triangle. b c N a a

10 224 Some triangle centers Exercise 1. onstruct the tritangent circles of a triangle. (1) Join each excenter to the midpoint of the corresponding side of. These three lines intersect at a point M i. (This is called the Mittenpunkt of the triangle). (2) Join each excenter to the point of tangency of the incircle with the corresponding side. These three lines are concurrent at another point T. (3) The lines M i and T are symmetric with respect to the bisector of angle ; so are the lines M i, T and M i, T (with respect to the bisectors of angles and ). b c T M i a

11 7.4 The excircles and the Nagel point onstruct the excircles of a triangle. (1) Let D, E, F be the midpoints of the sides,,. onstruct the incenter S p of triangle DEF, 5 and the tangents from S to each of the three excircles. (2) The 6 points of tangency are on a circle, which is orthogonal to each of the excircles. b c S p a 5 This is called the Spieker point of triangle.

12 226 Some triangle centers 3. Let D, E, F be the midpoints of the sides,,, and let the incircle touch these sides at,, respectively. The lines through parallel to D, through to E and through to F are concurrent. 6 F E P D 6 rux Math. Problem The reflection of the Nagel point N a in the incenter. This is 145 of ET.

13 7.5 the excentral triangle the excentral triangle The excentral triangle has vertices the excenters a, b, c. ts sides are the external bisectors of the angles of triangle, so that the internal bisectors are its altitudes, and its orthocenter. Thus, is the orthic triangle of a b c, and its circumcircle is the nine-point circle of the excentral triangle. Therefore, the line O is the Euler line of the excentral triangle. b M c O M a We also have the following interesting facts. (i) The midpoints of the segments a, b, and c are on the circumcircle of. (ii) The midpoints of b c, c a, and a b are also on the circumcircle of. (iii) n particular, the midpoints M of a and M of b c are antipodal on the circumcircle, and MM is the perpendicular bisector of. b c O M a More surprising are the following facts about the circumcirle of the excentral triangle. (iv) The circumradius of the excentral circle is 2R.

14 228 Some triangle centers (v) Since the excentral triangle has nine-point center O and orthocenter, its circumcenter is the reflection of in O, i.e., := 2O. Note that the line a is perpendicular to. t therefore contains the point of tangency with the -excircle. From this, we deduce two more interesting facts. Proposition 7.1. The perpendiculars from three excenters to the corresponding sides concur at the reflection of the incenter in the circumcenter. b M c O c D a b M a Theorem 7.2. r a + r b + r c = 4R + r. Proof. onsider the diameter MM of the circumcircle (of ) perpendicular to. (i) r b + r c = 2 DM = 2(R + OD). (ii) 2 OD = r + a = r + 2R r a. (iii) t follows that r a + r b + r c = 4R + r.

Example G1: Triangles with circumcenter on a median. Prove that if the circumcenter of a triangle lies on a median, the triangle either is isosceles

1 Example G1: Triangles with circumcenter on a median. Prove that if the circumcenter of a triangle lies on a median, the triangle either is isosceles or contains a right angle. D D 2 Solution to Example