L> trying out TrianglesConstructing Trianglesby Jennifer WhitmireLet"s start by contructing the centroid ofa triangle.The centroid (G) the a triangle is the commonintersection of the three medians the a triangle. A median of atriangle is the segment from a vertex to the midpoint of the oppositeside.The median divides the triangle into two areas.When we construct all 3 medians, we finish up with six internaltriangles.Now, let"s discover the orthocenter the a triangle.The orthocenter (H) the a triangle is the commonintersection that the three lines comprise the altitudes. One altitudeis a perpendicular segment native a vertex to the line of the oppositeside.Let"s build the circumcenter of a triangle.The circumcenter (C) the a triangle is the pointin the plane equidistant from the 3 vertices that the triangle.Since a suggest equidistant from 2 points lies on the perpendicularbisector that the segment established by two points, (C) is ~ above thethe perpendicular bisector of each side of the triangle. Note(C) might be exterior the triangle.Let"s look at the incenter the a triangle.The incenter (I) of the triangle is the pointon the interior of the triangle the is equidistant from every sides.Since a allude interior come an angle that is equidistant native thetwo sides lies top top the angle bisector, then (I) should be ~ above theangle bisector of each angle that the triangle.Now, let"s view if there is a connection betweenthe centroid, the orthocenter, the circumcenter, and the incenterof a triangle.When G,H,C, and I are built together,we can see the H,G, and C type a line. As soon as the triangle changesshape they still stay in a line.What if us look in ~ the centroid, orthocenter,circumcenter, and incenter that the medial triangle and compareit to the original triangle.The medial triangle was drawn by connectingthe midpoints that the original triangle. H,C,G, and also I were constructedon the medial triangle, and also h,c,g, and also i were created on theoriginal triangle. G and g were discovered to be the same allude meaningthat the centroid is the exact same for the initial triangle and also themedial triangle.What if us look at the orthic triangle andfind the centroid, orthocenter, circumcenter, and also incenter? Isthere a relationship between those the the orthic triangle andthe original triangle?This illustration is the the orthic triangle. Itwas built by connecting the feet of the altitudes of theoriginal triangle. Let"s look in ~ G,H,C, and I for the orthic triangleand conpare it come the original triangle.The H,G,C, and I all appear to continue to be in thesame directly line in the orthic triangle and the original triangle.What if we adjust the triangle come a appropriate triangle and an obtusetriangle. Let"s look and also see.For the best right triangle, it shows up thatthe points continue to be in their very same straight line splitting the triangleinto 2 equal areas.For the obtuse triangle, ours orthic triangleis gone. The circumcenter has actually moved exterior the triangle, andthe orthocenter has moved outside the triangle. The point out nolonger show up to be in a directly line. The brand seem confusing,so I will construct the point out again on this triangle.The points were correct, therefore I can assume thatH, G, and also C always remain in a line; however, the incenter doesnot show up on that line.Now let"s build the the three secondarytriangles, the medial triangle, the orthic triangle, and also the trianglearound the orthocenter.Take the acute triangle ABC. Triangle GHI isthe medial triangle, and also triangle DEF is a triangle constructedby illustration HA, HB, and HC and constructing a triangle through connectingtheir midpoints. How have the right to we prove the triangle DEF is similarto triangle ABC and also congruent to triangle GHI?First let"s find the G,H,C, and also I for eachtriangle.The G,H, and C stayed in the same line in allthree triangles. Plenty of of the points to be the same for 2 of thetriangles.When the circumcircle of every of the threesecondary triangle is constructed, they room the exact same circle.The red triangle was constructed by connecting the midpoints ofthe political parties of the initial triangle. The blue triangle to be constructedby finding the orthocenter that the original triangle, connectingthe orthocenter come the three vertices, detect the midpoint ofeach side, and then connecting those three midpoints. The circumcirclewas constructed around each the the three secondary triangles,and all three circumcircles were the exact same circle i m sorry is difficultto watch on one construction.Let"s look at the nine suggest circle. It isconstructed v the 3 midpoints that the sides, the threefeet that the altitudes, and also the three midpoints that the segmentsconnecting the vertices to the orthocenter. (N) is the centerof the nine allude circle.Let"s see how (N) is related to (G), (H), (C),and (I).It appears as though (N) is top top the same lineas (G),(H), (C), and also (I). Let"s view if it still on the exact same linewhen the triangle alters shape.The center of the nine allude circle remainson the same line together the orthocenter, circumcenter, and also centroidof the circle.Let"s look in ~ the three perpendicular bisectorsof a triangle.DG, EH, and IF room perpendicular bisectors.The perpendicular bisectors the a triangle are perpendicular linesthrough the midpoints of each side.Let"s look at the 3 altitudes the a triangle.DC, FB, and also EA are the 3 altitudes that thetriangle. The 3 altitudes of a triangle room perpendicularlines indigenous the verteces come the the contrary sides.Let"s look in ~ the three medians.The 3 medians are lines connecting themidpoints of the political parties to opposing vertex.Now, let"s look in ~ the 3 angle bisectors.Can we prove that the 3 angle bisectorsof the internal angles that a triangle space concurrent?CE is an angle bisector of edge ACD, and DEis an angle bisector of angle CDA. Because allude F is ~ above bothangle bisectors and is equidistant from each side of the triangle,it is additionally on AC i beg your pardon is the angle bisector of edge DAC.
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Sincethe edge bisector is equidistant from all 3 sides, the wouldhave to it is in the same point. Because of this the 3 angle bisectorsof a triangle space concurrent.Take the suggest of concurrence and also constructa circle tangent come all 3 lines.We can see the this circle tangent come allthree sides has actually points the tangency in ~ the intersection the eachangle bisector and the opposite side.RETURN