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In mathematics, what coincidences can make people shine?
As we all know, every triangle has a circumscribed circle and an inscribed circle. Their centers are called external heart and internal heart respectively. The outer center is the intersection of three perpendicular lines of the triangle, and the inner center is the intersection of three bisectors of the inner angle. This is perhaps the simplest and most well-known coincidence in plane geometry. However, this property is generally wrong for quadrangles. Most quadrangles have neither circumscribed circle nor inscribed circle. A circle that passes through three vertices does not necessarily pass through the fourth vertex, and a circle that is tangent to three sides is not necessarily tangent to the fourth side. However, there are always some lucky quadrangles, some of which are circumscribed and some are inscribed. These lucky people also have excellent attributes that ordinary quadrangles do not have. For example, if a quadrilateral has an inscribed circle, then the tangent points of its diagonal and opposite sides are connected by four lines. This is a beautiful coincidence. And the name of this theorem is Newton's theorem. Yes, Newton discovered gravity.