# Thread:

1. ## Diagonal

Consider a convex n-gon such that no 3 diagonals intersect at a single point. Draw all the diagonals (i.e. connect every pair of vertices by a segment).

a) How many intersections do the diagonals determine?

b) Into how many parts is the polygon divided by the diagonals?

2. Originally Posted by vivian
Consider a convex n-gon such that no 3 diagonals intersect at a single point. Draw all the diagonals (i.e. connect every pair of vertices by a segment).

a) How many intersections do the diagonals determine?

b) Into how many parts is the polygon divided by the diagonals?
Hello,

to a). from each vertex there are starting (n-3) diagonals.

The total number of diagonals is therefore: $\frac n2 \cdot (n-3)$

Each diagonal is intersected by (n-3) diagonals. Therefore the number of intersections is: $\frac n2 \cdot (n-3)^2$

to b) I'm not quite sure what you mean by parts
Attached Thumbnails
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 2, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9232874512672424, "perplexity_flag": "middle"}