## Thursday, May 26, 2016

### Geometry Problem 1218: Scalene Triangle, Equilateral Triangles, Midpoints, 60 Degrees, Congruence

Geometry Problem. Post your solution in the comment box below.
Level: Mathematics Education, High School, Honors Geometry, College.

Click the figure below to view more details of problem 1218.

1. http://s33.postimg.org/53zo0aqen/pro_1218.png
Let D is the midpoint of BC
CBC2 is 30-60-90 triangle
So BC2=1/2. BC= BD , BDC2 is equilateral and ∡ (CDC2)= 120 degrees.
Note that DB1= ½. AB= BA2
∡ (BDC2)= ∡ (B1DC)+ 120 = ∡ (ABC)+120= ∡ (A2BC2)
Triangle B1DC2 congruent to tri. A2BC2 ( case SAS)
So B1C2= A2C2 and ∡ (B1C2A2)= ∡ (DC2B)= 60 degrees
So B1C2A2 is a equilateral triangle.

2. Problem 1218

Let K, L medpoints the sides AB and AC.Then KA2=AB/2=A2B=KB=LB1, KB1=BL=LC2, <B1KA2=<ΑΚΒ1+120=<ABC+20=<B1LC+120=B1LC2=<ABC+60+60=120. Therefore
Triangles B1KA2, A2BC2 and B1LC2 are equals.Then triangle A2B1C2 is equilateral.
MANOLOUDIS APOSTOLIS
4th HIGH SCHOOL OF KORYDALLOS -PIRAEUS-GREECE

3. Each of the sides of the triangle are = to a/2 + c/2 using the mid point theorem and so the result follows

Sumith Peiris
Moratuwa
Sri Lanka

4. Refer to your comment " Each of the sides of the triangle are = to a/2 + c/2".
Per sketch shown on the link
http://s33.postimg.org/53zo0aqen/pro_1218.png
each side of triangle B1C2A2 is less than a/2 + c/2 .