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Discussion: Incircles
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%3 N_20240702211643b_ColinWright    By ColinWright 2024/07/02 @ 21:16:43b -------------------------------- For this family we can show the radius is always k.        (select only this node)     N_20240702212141a_ColinWright    By ColinWright 2024/07/02 @ 21:21:41a -------------------------------- But there are other PPTs, so some integers will occur more than once.        (select only this node)     N_20240702211643b_ColinWright->N_20240702212141a_ColinWright N_20240702212259a_ColinWright    By ColinWright 2024/07/02 @ 21:22:59a -------------------------------- Radius = 3, for example.        (select only this node)     N_20240702212141a_ColinWright->N_20240702212259a_ColinWright N_20240702212259b_ColinWright    By ColinWright 2024/07/02 @ 21:22:59b -------------------------------- Turns up in 7:24:25 and in 8:15:17.        (select only this node)     N_20240702212259c_ColinWright    By ColinWright 2024/07/02 @ 21:22:59c -------------------------------- For a given radius, can we compute how many PPTs have that inradius?        (select only this node)     N_20240702212259b_ColinWright->N_20240702212259c_ColinWright N_20240702212259d_ColinWright    By ColinWright 2024/07/02 @ 21:22:59d -------------------------------- Yes.        (select only this node)     N_20240702212259c_ColinWright->N_20240702212259d_ColinWright N_20240702213747a_ColinWright    By ColinWright 2024/07/02 @ 21:37:47a -------------------------------- It's a sub-family ... there are others.        (select only this node)     N_20240630182248k_ColinWright    By ColinWright 2024/06/30 @ 18:22:48k -------------------------------- 8:15:17 -> Radius 3        (select only this node)     N_20240702213747a_ColinWright->N_20240630182248k_ColinWright N_20240630182248k_ColinWright->N_20240702212259a_ColinWright N_20240630182248l_ColinWright    By ColinWright 2024/06/30 @ 18:22:48l -------------------------------- Is every radius an integer?        (select only this node)     N_20240630182248k_ColinWright->N_20240630182248l_ColinWright N_20240630182248m_ColinWright    By ColinWright 2024/06/30 @ 18:22:48m -------------------------------- Does every positive integer occur?        (select only this node)     N_20240630182248k_ColinWright->N_20240630182248m_ColinWright N_20240702212259a_ColinWright->N_20240702212259b_ColinWright N_20240702214250a_ColinWright    By ColinWright 2024/07/02 @ 21:42:50a -------------------------------- Example        (select only this node)     N_20240630182248c_ColinWright    By ColinWright 2024/06/30 @ 18:22:48c -------------------------------- 3:4:5 Triangle        (select only this node)     N_20240702214250a_ColinWright->N_20240630182248c_ColinWright N_20240630182248j_ColinWright    By ColinWright 2024/06/30 @ 18:22:48j -------------------------------- 7:24:25 -> Radius 3        (select only this node)     N_20240702214250a_ColinWright->N_20240630182248j_ColinWright N_20240630182248f_ColinWright    By ColinWright 2024/06/30 @ 18:22:48f -------------------------------- Next in the family ... 5:12:13 ... what is the radius of the incircle?        (select only this node)     N_20240702214250a_ColinWright->N_20240630182248f_ColinWright N_20240630182248a_ColinWright    By ColinWright 2024/06/30 @ 18:22:48a -------------------------------- Incircles and Triangles        (select only this node)     N_20240630182248b_ColinWright    By ColinWright 2024/06/30 @ 18:22:48b -------------------------------- Primitive Pythagorean Triangles        (select only this node)     N_20240630182248a_ColinWright->N_20240630182248b_ColinWright N_20240630182248d_ColinWright    By ColinWright 2024/06/30 @ 18:22:48d -------------------------------- What is the radius of the incircle?        (select only this node)     N_20240630182248c_ColinWright->N_20240630182248d_ColinWright N_20240630182248c_ColinWright->N_20240630182248f_ColinWright N_20240630182248e_ColinWright    By ColinWright 2024/06/30 @ 18:22:48e -------------------------------- Exactly 1.        (select only this node)     N_20240630182248d_ColinWright->N_20240630182248e_ColinWright N_20240630182248e_ColinWright->N_20240630182248l_ColinWright N_20240630182248h_ColinWright    By ColinWright 2024/06/30 @ 18:22:48h -------------------------------- Generate the family        (select only this node)     N_20240630182248f_ColinWright->N_20240630182248h_ColinWright N_20240630182248g_ColinWright    By ColinWright 2024/06/30 @ 18:22:48g -------------------------------- Exactly 2.        (select only this node)     N_20240630182248f_ColinWright->N_20240630182248g_ColinWright N_20240630182248g_ColinWright->N_20240630182248l_ColinWright N_20240630182248i_ColinWright    By ColinWright 2024/06/30 @ 18:22:48i -------------------------------- 2k+1 : 2k^2+2k : 2k^2+2k+1        (select only this node)     N_20240630182248h_ColinWright->N_20240630182248i_ColinWright N_20240702211643a_ColinWright    By ColinWright 2024/07/02 @ 21:16:43a -------------------------------- Yes.        (select only this node)     N_20240630182248m_ColinWright->N_20240702211643a_ColinWright N_20240630182248q_ColinWright    By ColinWright 2024/06/30 @ 18:22:48q -------------------------------- Area of any triangle in terms of r and s        (select only this node)     N_20240630182248r_ColinWright    By ColinWright 2024/06/30 @ 18:22:48r -------------------------------- Accidentally prove Pythagoras        (select only this node)     N_20240630182248q_ColinWright->N_20240630182248r_ColinWright N_20240630182248s_ColinWright    By ColinWright 2024/06/30 @ 18:22:48s -------------------------------- Emergence of the semi-perimeter        (select only this node)     N_20240630182248r_ColinWright->N_20240630182248s_ColinWright N_20240702210731a_ColinWright    By ColinWright 2024/07/02 @ 21:07:31a -------------------------------- Definition.        (select only this node)     N_20240630182248b_ColinWright->N_20240702210731a_ColinWright N_20240630182248b_ColinWright->N_20240630182248c_ColinWright N_20240630182248n_ColinWright    By ColinWright 2024/06/30 @ 18:22:48n -------------------------------- We have a formula for the sub-family        (select only this node)     N_20240630182248i_ColinWright->N_20240630182248n_ColinWright N_20240630182248i_ColinWright->N_20240702211643a_ColinWright N_20240630182248i_ColinWright->N_20240630182248j_ColinWright N_20240630182248i_ColinWright->N_20240702213747a_ColinWright N_20240630182248j_ColinWright->N_20240630182248l_ColinWright N_20240630182248j_ColinWright->N_20240702212259a_ColinWright N_20240702211328a_ColinWright    By ColinWright 2024/07/02 @ 21:13:28a -------------------------------- Yes.        (select only this node)     N_20240630182248l_ColinWright->N_20240702211328a_ColinWright N_20240630182248o_ColinWright    By ColinWright 2024/06/30 @ 18:22:48o -------------------------------- Is there are formula for all PPT?        (select only this node)     N_20240630182248n_ColinWright->N_20240630182248o_ColinWright N_20240630182248p_ColinWright    By ColinWright 2024/06/30 @ 18:22:48p -------------------------------- c^2-d^2 : 2cd : c^2+d^s where gcd(c,d)=1 and not both odd.        (select only this node)     N_20240630182248o_ColinWright->N_20240630182248p_ColinWright N_20240630182248p_ColinWright->N_20240630182248q_ColinWright N_20240702210731a_ColinWright->N_20240702214250a_ColinWright N_20240702211328b_ColinWright    By ColinWright 2024/07/02 @ 21:13:28b -------------------------------- We can show the radius is (a+b-c)/2, exactly one of a, b, and c is even, hence (a+b-c) is in 2Z.        (select only this node)     N_20240702211328a_ColinWright->N_20240702211328b_ColinWright N_20240702211328a_ColinWright->N_20240630182248m_ColinWright N_20240702211328b_ColinWright->N_20240702211643a_ColinWright N_20240702211643a_ColinWright->N_20240702211643b_ColinWright