Cevians, Symmedians, and Excircles. MA 341 Topics in Geometry Lecture 16


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1 Cevians, Symmedians, and Excircles MA 341 Topics in Geometry Lecture 16
2 Cevian A cevian is a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension). B cevian A D C 05Oct2011 MA
3 Cevian Triangle & Circle Pick P in the interior of ABC Draw cevians from each vertex through h P to the opposite side Gives set of three intersecting cevians AA, BB, and CC with respect to that point. The triangle A B C ABC is known as the cevian triangle of ABC with respect to P Circumcircle of A B C is known as the evian circle with respect to P. 05Oct2011 MA
4 Cevian circle Cevian triangle 05Oct2011 MA
5 Cevians In ABC examples of cevians are: medians cevian point = G perpendicular bisectors cevian point = O angle bisectors cevian point = I (incenter) altitudes cevian point = H Ceva s Theorem deals with concurrence of any set of cevians. 05Oct2011 MA
6 Gergonne Point In ABC find the incircle and points of tangency of incirclei with ih sides of ABC. Known as contact triangle 05Oct2011 MA
7 Gergonne Point These cevians are concurrent! Why? Recall that AE=AF, BD=BF, and CD=CE Ge 05Oct2011 MA
8 Gergonne Point The point is called the Gergonne point, Ge. Ge 05Oct2011 MA
9 Gergonne Point Draw lines parallel to sides of contact triangle through Ge. 05Oct2011 MA
10 Gergonne Point Six points are concyclic!! Called the Adams Circle 05Oct2011 MA
11 Gergonne Point Center of Adams circle = incenter of ABC 05Oct2011 MA
12 Isogonal Conjugates Two lines AB and AC through vertex A are said to be isogonal if one is the reflection of the other through the angle bisector. 05Oct2011 MA
13 Isogonal Conjugates If lines through A, B, and C are concurrent at P, then the isogonal lines are concurrent at Q. Points P and Q are isogonal conjugates. 05Oct2011 MA
14 Symmedians In ABC, the symmedian AS a is a cevian through vertex A (S a BC) isogonally conjugate to the median AM a, M a being the midpoint of BC. The other two symmedians BS b and CS c are defined similarly. 05Oct2011 MA
15 Symmedians The three symmedians AS a, BS b and CS c concur in a point commonly denoted K and variably known as either the symmedian point or the Lemoine point 05Oct2011 MA
16 Symmedian of Right Triangle The symmedian point K of a right triangle is the midpoint of the altitude to the hypotenuse. A K M b B D C 05Oct2011 MA
17 Proportions of the Symmedian Draw the cevian from vertex A, through the symmedian point, to the opposite side of the triangle, meeting BC at S a. Then c b BS 2 a c 2 CS b a a 05Oct2011 MA
18 Length of the Symmedian Draw the cevian from vertex C, through the symmedian point, to the opposite side of the triangle. Then this segment has length Likewise ab 2a 2b c CS c 2 2 a b bc 2b 2c a AS a 2 2 b c ac 2a 2c b BS b 2 2 a c Oct2011 MA
19 Excircles In several versions of geometry triangles are defined in terms of lines not segments. A B C 05Oct2011 MA
20 Excircles Do these sets of three lines define circles? Known as tritangent circles A B C 05Oct2011 MA
21 Excircles I C B r c A I r b I B C I A r a 05Oct2011 MA
22 Construction of Excircles 05Oct2011 MA
23 Extend the sides 05Oct2011 MA
24 Bisect exterior angle at A 05Oct2011 MA
25 Bisect exterior angle at B 05Oct2011 MA
26 Find intersection I c 05Oct2011 MA
27 Drop perpendicular to AB I c 05Oct2011 MA
28 Find point of intersection with AB I c 05Oct2011 MA
29 Construct circle centered at I c I c r c 05Oct2011 MA
30 05Oct2011 MA
31 Excircles The I a, I b, and I c are called excenters. r a, r b, r c are called exradii 05Oct2011 MA
32 Excircles Theorem: The length of the tangent from a vertex to the opposite exscribed circle equals the semiperimeter, s. CP = s 05Oct2011 MA
33 Excircles 1. CQ = CP 2. AP = AY 3. CP = CA+AP = CA+AY 4. CQ= BC+BY 5. CP + CQ = AC + AY + BY + BC 6. 2CP = AB + BC + AC = 2s 7. CP = s 05Oct2011 MA
34 Exradii 1. CP I C P 2. tan(c/2)=r C /s 3. Use Law of Tangents I c C (s a)(s b) s(s a)(s b) r stan s c 2 s(s c) s c 05Oct2011 MA
35 Exradii Likewise r a r b r c s(s b)(s c) s a s(s a)(s c) s b s(s a)(s b) s c 05Oct2011 MA
36 Excircles Theorem: For any triangle ABC r r r r a b c 05Oct2011 MA
37 Excircles s a s b s c r r r s(s b)(s c) s(s a)(s c) s(s a)(s b) a b c s a s b s c s(s a)(s b)(s c) s(s a)(s b)(s c) s(s a)(s b)(s c) 3s (a b c) s(s a)(s b)(s c) s s(s a)(s b)(s c) s K 1 r 05Oct2011 MA
38 Nagel Point In ABC find the excircles and points of tangency of the excircles with ih sides of ABC. 05Oct2011 MA
39 Nagel Point These cevians are concurrent! 05Oct2011 MA
40 Nagel Point Point is known as the Nagel point 05Oct2011 MA
41 Mittenpunkt Point The mittenpunkt of ABC is the symmedian point of the excentral triangle ( I a I b I c formed from centers of excircles) 05Oct2011 MA
42 Mittenpunkt Point The mittenpunkt of ABC is the point of intersection of the lines from the excenters through midpoints of corresponding sides 05Oct2011 MA
43 Spieker Point The Spieker center is center of Spieker circle, i.e.,., the incenter of the medial triangle of the original triangle. 05Oct2011 MA
44 Special Segments Gergonne point, centroid and mittenpunkt are collinear GGe =2 GM 05Oct2011 MA
45 Special Segments Mittenpunkt, Spieker center and orthocenter are collinear 05Oct2011 MA
46 Special Segments Mittenpunkt, incenter and symmedian point K are collinear with distance ratio IM 2(a +b + c ) = 2 MK (a +b +c) 05Oct2011 MA
47 Nagel Line The Nagel line is the line on which the incenter, triangle centroid, Spieker center Sp, and Nagel point Na lie. GNa =2 IG 05Oct2011 MA
48 Various Centers 05Oct2011 MA
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