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wu :: forums    riddles    medium (Moderators: Grimbal, Icarus, Eigenray, ThudnBlunder, william wu, towr, SMQ)    integral inradius and circumradius « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: integral inradius and circumradius  (Read 1711 times) Christine Full Member     Posts: 159 integral inradius and circumradius   « on: Nov 20th, 2011, 12:43pm » Quote Modify (1) For each integer n it is possible to construct an integer triangle whose inradius is n? How?   (2) Can we always have integral circumradius as well? IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: integral inradius and circumradius   « Reply #1 on: Nov 20th, 2011, 12:54pm » Quote Modify 1) the triangle 3-4-5 has inradius 1, so any K-multiple would have radius K   For 2) do you mean at the same time as having an integer triangle with integer inradius? IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Christine Full Member     Posts: 159 Re: integral inradius and circumradius   « Reply #2 on: Nov 20th, 2011, 1:06pm » Quote Modify Quote:   For 2) do you mean at the same time as having an integer triangle with integer inradius?   We may ask   (a) integral sides + integral circumradius (b)  integral sides +  integer inradius + integer circumradius IP Logged Christine Full Member     Posts: 159 Re: integral inradius and circumradius   « Reply #3 on: Nov 20th, 2011, 1:13pm » Quote Modify I just saw that it only works with Pythagorean triangle   I found these 2 theorems:   The inradius of a Pythagorean triangle is an integer.   The inradius of a Heronian triangle is a rational number.   But I don't know whether we always have integral circumradius as well (either at the same time or not)? « Last Edit: Nov 20th, 2011, 1:15pm by Christine » IP Logged Pages: 1  Reply Notify of replies Send Topic Print Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

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