SECTION 1.2 Triangle Geometry 19
- In the figure to the right, three cir-
cies of the same radius and centers
X, Y andZare shown intersecting
at pointsA, B, C, and D, with D
the common point of intersection of
all three circles.
Show that
(a) D is the circumcenter of
4 XY Z, and that
(b)Dis the orthocenter of 4 ABC.
(Hint: note thatY ZCDis
a rhombus.) - Show that the three medians of a triangle divide the triangle into
six triangle of equal area. - Let the triangle 4 ABC be given, and letA′ be the midpoint of
[BC],B′the midpoint of [AC] and letC′be the midpoint of [AB].
Prove that
(i) 4 A′B′C′∼ 4ABC and that the ratios of the corresponding
sides are 1:2.
(ii) 4 A′B′C′and 4 ABC have the same centroid.
(iii) The four triangles determined within 4 ABC by 4 A′B′C′
are all congruent.
(iv) The circumcenter of 4 ABC is the orthocenter of 4 A′B′C′.The triangle 4 A′B′C′ of 4 ABC formed above is called theme-
dial triangleof 4 ABC.- The figure below depicts a hexagram “inscribed” in two lines. Us-
ing the prompts given, show that the linesX, Y, andZare colin-
ear. This result is usually referred toPappus’ theorem.