JSFH LUkld2l0aEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYjSSlHcm9lYm5lckdGJA== 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 The Circle Theorem of Apollonius 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QyQtSSZCYXNpc0c2IjYkSSNIMUdGJS1JJXRkZWdHRiU2Ji1JJHNlcUclKnByb3RlY3RlZEc2JCZJInhHRiU2I0kiaUdGJS9GMjsiIiIiIikmSSJ1R0YlNiNGNSZGODYjIiIjSSJ5R0YlRjU= NykmSSJ1RzYiNiMiIiMmRiQ2IyIiIiZJInhHRiU2IyIiJSZGLDYjIiIkJkYsRiYmRixGKSwsKiZJInlHRiVGKiZGLDYjIiImRidGKkYqRioqJkY2RiomRiw2IyIiJ0YnRioqKEY2RipGN0YqJkYsNiMiIihGKiEiIyooRjZGKkY7RiomRiw2IyIiKUYqRkI= QyQ+SSNIMkc2IjwjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJkkiaEdGJTYjSSJpR0YlL0YuOyIiIiIiKUYx PCosJiZJInhHNiI2IyIiIiIiIyZJInVHRiZGJyEiIiwmJkYlNiNGKUYpJkYrRi9GLCwmJkYlNiMiIiRGKUYqRiwsJiZGJTYjIiIlRilGMEYsLCYqJiZGJTYjIiImRihGKkYoRigqJiZGJTYjIiInRihGMEYoRiwsKComRjtGKEYwRihGKComRj9GKEYqRihGKComRipGKEYwRihGLCwqKiQsJkYkRigmRiU2IyIiKEYsRilGKCokJkYlNiMiIilGKUYoKiRGSUYpRiwqJCwmRi5GKEZNRixGKUYsLCpGR0YoRkxGKCokLCZGMkYoRklGLEYpRiwqJCwmRjZGKEZNRixGKUYs LUkrTm9ybWFsRm9ybUc2IjYmSSJnR0YkSSNIMkdGJC1JJXRkZWdHRiQ2JS1JJHNlcUclKnByb3RlY3RlZEc2JCZJInhHRiQ2I0kiaUdGJC9GMjsiIiIiIikmSSJ1R0YkNiNGNSZGODYjIiIjLkkiUUdGJA== LC4qJiZJInhHNiI2IyIiJyIiIiZGJTYjIiIpRikiIiMqJCZJInVHRiY2I0YpRi0jRikiIiUqJCZGJTYjIiImRi0hIiIqJkY1RikmRiU2IyIiKEYpRi0qJEYkRi1GOComRi9GKUY6RilGOA== QyRJIlFHNiIiIiI= NyosKCZJInhHNiI2IyIiIiNGKCIiIyZGJTYjIiIoISIiJkkidUdGJkYnI0YoIiIlIiIhRjNGM0YzRjNGM0Yz Groebner basis in the ring k(u1, u2)[x1,...,x8]: 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 NyosJiZJInhHNiI2IyIiKSIiJSZJInVHRiY2IyIiIyEiIiwmJkYrNiMiIiJGLiZGJTYjIiIoRiksJiomRjBGLUYqRjJGLiomLCYqJEYwRi1GMiokRipGLUYyRjImRiU2IyIiJ0YyRjIsJiomRjBGMkYqRi1GLiomRjlGMiZGJTYjIiImRjJGMiwmJkYlNiNGKUYtRipGLiwmJkYlNiMiIiRGLUYwRi4sJiZGJUYsRi1GKkYuLCYmRiVGMUYtRjBGLg== LUkrTm9ybWFsRm9ybUc2IjYmSSJnR0YkLUkoY29udmVydEclKnByb3RlY3RlZEc2JEkiQkdGJEkkc2V0R0YpLUkldGRlZ0dGJDYjLUkkc2VxR0YpNiQmSSJ4R0YkNiNJImlHRiQvRjY7IiIiIiIpLkkiUUdGJA== IiIh The Parallelogram Theorem Qyo+JkkiaEc2IjYjIiIiLCYmSSJ4R0YmNiMiIiNGKCZJInVHRiY2IyIiJCEiIkYoPiZGJUYsLCYqJiwmJkYrRidGKCZGL0YnRjJGKEYuRihGKComJkYvRixGKEYuRihGMkYoPiZGJUYwLCYqJiZGKzYjIiIlRihGOEYoRigqJkYuRigmRitGMEYoRjJGKD4mRiVGQSwmKiZGQEYoLCZGO0YoRjlGMkYoRigqJiwmRkRGKEY5RjJGKEYuRihGMkYo LCYmSSJ4RzYiNiMiIiMiIiImSSJ1R0YlNiMiIiQhIiI= LCYqJiwmJkkieEc2IjYjIiIiRikmSSJ1R0YnRighIiJGKSZGKzYjIiIkRilGKSomJkYrNiMiIiNGKUYtRilGLA== LCYqJiZJInhHNiI2IyIiJSIiIiZGJTYjRilGKUYpKiYmSSJ1R0YmNiMiIiRGKSZGJUYvRikhIiI= LCYqJiZJInhHNiI2IyIiJSIiIiwmJkkidUdGJjYjIiIjRikmRiw2I0YpISIiRilGKSomLCYmRiU2IyIiJEYpRi9GMUYpJkYsRjVGKUYx QyQ+SSJnRzYiLCoqJCZJInhHRiU2IyIiJCIiIyIiIiokJkYpNiMiIiVGLEYtKiQsJkYoRi0mRik2I0YtISIiRixGNiokLCZGL0YtJkYpNiNGLEY2RixGNkYt LCoqJCZJInhHNiI2IyIiJCIiIyIiIiokJkYlNiMiIiVGKUYqKiQsJkYkRiomRiU2I0YqISIiRilGMyokLCZGLEYqJkYlNiNGKUYzRilGMw== QyQ+SSNIMUc2IjwkLCYiIiJGKComSSJ5R0YlRihJImdHRiVGKCEiIi1JJHNlcUclKnByb3RlY3RlZEc2JCZJImhHRiU2I0kiaUdGJS9GNDtGKCIiJUYo PCcsJiIiIkYkKiZJInlHNiJGJCwqKiQmSSJ4R0YnNiMiIiQiIiNGJCokJkYrNiMiIiVGLkYkKiQsJkYqRiQmRis2I0YkISIiRi5GNyokLCZGMEYkJkYrNiNGLkY3Ri5GN0YkRjcsJkY6RiQmSSJ1R0YnRixGNywmKiZGMEYkRjVGJEYkKiZGPUYkRipGJEY3LCYqJkYwRiQsJiZGPkY7RiQmRj5GNkY3RiRGJComLCZGKkYkRkZGN0YkRj1GJEY3LCYqJiwmRjVGJEZGRjdGJEY9RiRGJComRkVGJEY9RiRGNw== QyQtSSZCYXNpc0c2IjYkSSNIMUdGJS1JJXRkZWdHRiU2Jy1JJHNlcUclKnByb3RlY3RlZEc2JCZJInhHRiU2I0kiaUdGJS9GMjsiIiIiIiUmSSJ1R0YlNiNGNSZGODYjIiIjJkY4NiMiIiRJInlHRiVGNQ== 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QyQtSTNQcmltZURlY29tcG9zaXRpb25HNiI2I0kiSkdGJSIiIg== Warning, computation interrupted Qyo+JkkjaGhHNiI2IyIiIiwoJkkieEdGJkYnRigmSSJ1R0YmRichIiImRi02IyIiI0YuRig+JkYlRjAsJiZGK0YwRigmRi02IyIiJEYuRig+JkYlRjcsKCZGK0Y3RjFGLEYuRi9GLkYoPiZGJTYjIiIlLCYmRitGP0YxRjZGLkYo LCgmSSJ4RzYiNiMiIiJGJyZJInVHRiVGJiEiIiZGKTYjIiIjRio= LCYmSSJ4RzYiNiMiIiMiIiImSSJ1R0YlNiMiIiQhIiI= LCgmSSJ4RzYiNiMiIiQiIiMmSSJ1R0YlNiMiIiIhIiImRio2I0YoRi0= LCYmSSJ4RzYiNiMiIiUiIiMmSSJ1R0YlNiMiIiQhIiI= QyQ+SSRISDFHNiI8JCwmIiIiRigqJkkieUdGJUYoSSJnR0YlRighIiItSSRzZXFHJSpwcm90ZWN0ZWRHNiQmSSNoaEdGJTYjSSJpR0YlL0Y0O0YoIiIlRig= PCcsJiIiIkYkKiZJInlHNiJGJCwqKiQmSSJ4R0YnNiMiIiQiIiNGJCokJkYrNiMiIiVGLkYkKiQsJkYqRiQmRis2I0YkISIiRi5GNyokLCZGMEYkJkYrNiNGLkY3Ri5GN0YkRjcsJkY6RiQmSSJ1R0YnRixGNywmRjBGLkY9RjcsKEY1RiQmRj5GNkY3JkY+RjtGNywoRipGLkZBRjdGQkY3 QyQtSSZCYXNpc0c2IjYkSSRISDFHRiUtSSV0ZGVnR0YlNictSSRzZXFHJSpwcm90ZWN0ZWRHNiQmSSJ4R0YlNiNJImlHRiUvRjI7IiIiIiIlJkkidUdGJTYjRjUmRjg2IyIiIyZGODYjIiIkSSJ5R0YlRjU= NyMiIiI= QyQ+SSRISDJHNiI8Iy1JJHNlcUclKnByb3RlY3RlZEc2JCZJI2hoR0YlNiNJImlHRiUvRi47IiIiIiIlRjE= PCYsJiZJInhHNiI2IyIiIyIiIiZJInVHRiY2IyIiJCEiIiwmJkYlNiMiIiVGKEYqRi4sKCZGJTYjRilGKSZGK0Y1Ri4mRitGJ0YuLCgmRiVGLEYoRjZGLkY3Ri4= LUkrTm9ybWFsRm9ybUc2IjYmSSJnR0YkSSRISDJHRiQtSSV0ZGVnR0YkNiYtSSRzZXFHJSpwcm90ZWN0ZWRHNiQmSSJ4R0YkNiNJImlHRiQvRjI7IiIiIiIlJkkidUdGJDYjRjUmRjg2IyIiIyZGODYjIiIkLkkiUUdGJA== IiIh QyRJIlFHNiIiIiI= NyYsKCZJInhHNiI2IyIiIyEiIiZGJTYjIiIlRigmSSJ1R0YmNiMiIiRGKUYtLComRiVGL0YoJkYlNiMiIiJGKSZGLkY0RikmRi5GJ0YpLCZGNkY1RjdGNQ== QyQtSSZCYXNpc0c2IjYkSSNIMUdGJS1JJXRkZWdHRiU2JC1JJHNlcUclKnByb3RlY3RlZEc2JCZJInhHRiU2I0kiaUdGJS9GMjsiIiIiIiVJInlHRiVGNQ== NyMiIiI= JSFH JSFH