# The Problem

### A pentagram, which is a five-pointed star, is inscribed in
a regular pentagon. How many

triangles are formed?

###

### (Source: Mathematics Teaching in the Middle School)

# The Solution

### There are 35 triangles formed when a pentogram is inscribed
in a regular pentagon. If we label the outer pentagon using letters
A,B,C,D,E then we can label the pentagon formed by the interior
of the pentagram as G,H,I,J,K. Where the points A,G,I, and D are
on one line, and A,H,F,C are on another, while B,H,G,E is a third
line and C,J,I,E forms a fourth line through the outer pentagon,
forming the boundaries of the inner pentagon. From here we can
find the triangles listed below as we systematically identify
each given triangle:

##### ADE

##### ABD

##### BCD

##### ABC

##### ABF

##### BCF

##### CDF

##### ADF

##### ABE

##### ABH

##### ABG

##### AHG

##### AEG

##### EGD

##### BFH

##### CEH

##### EGI

##### CFJ

##### DEI

##### DEJ

##### CDJ

##### CDI

##### CDE

##### DIJ

##### EBD

##### ACD

##### BEJ

##### ACI

##### BCE

##### ACE

##### AEI

##### BCJ

##### CDF

##### DEG

##### ADE