Aim: What is Ceva’s Theorem?

Aim: What is Ceva’s Theorem? Orthocenter Centroid Circumcenter Incircle a c d b But before that…a quick review! Perpendicular Bisectors Angle bisectors Medians Altitudes

Giovanni Ceva (1647-1734) • An Italian professor who taught mathematics in Mantua, working on geometry for most of his life. • He published a theorem in 1678 in De Lineis Rectis, Latin for “The Straight Lines” • His first major work, as well as his most important, De Lineis Rectis proved many geometrical propositions, most famously Ceva’s Theorem. It focused on the center of gravity between a system of points to obtain the relations of the segments.

Ceva’s Theorem • This theorem states that if three lines are drawn in a triangle from each vertex to the opposite sides, they intersect in a single point if, and only if, the sides are divided into parts so that:AB’ . CA’ . BC’ B’C A’B C’A = 1 (This theorem can also be written as AB’  CA’  BC’ = B’C  A’B  C’A)

According to Ceva’s Theorem, the lines, known as cevians, can intersect either inside or outside the triangle. However, the given equations only apply when the cevians intersect inside the triangle.The cevians MUST be concurrent to make the equation equal 1.

Cevians • In Giovanni’s honor, the intersecting lines, or AA’, BB’, and CC’ are known as cevians. • Although an unfamiliar term, cevians are a simple concept. As they are simply lines in a triangle that are concurrent, or intersecting in the same point, medians, altitudes, angle bisectors, and perpendicular bisectors of a triangle are all cevians.

Practice with Ceva’s theorem 1. 2. 3. AE = 8FB = 2CD = 3EC = 6AF = 2DB =? FB = 8AF = 10EC = 4DB = 4 AE = 4 CD = ? AF = 6 FB = 5EC = 2 AE = 8 CD = 3DB = ? DB = 4 CD = 5 DB = 10

Work 1. AE X CD X FB = EC X DB X FA 8 X 3 X 2 = 6 X DB X 2 8 X 6 = 12 DB 48 = 12 DB 4 = DB 2. AE X CD X FB = EC X DB X FA 4 X CD X 8 = 4 X 4 X 10 32 CD = 16 X 10 32 CD = 160 CD = 5 3. AE X CD X FB = EC X DB X FA 8 X 3 X 5 = 2 X DB X 6 24 X 5 = 12 DB 120 = 12 DB 10 = BD

Other terms • Cevian triangle: The triangle formed by the points on the concurrent lines opposite the vertices. For example, in the given triangle, the Cevian triangle for P is A’B’C’ • Anticevian: By extending the cevians past the original triangle, the cevian triangle forms outside the original triangle, forming the anticevian.

Ceva’s Conjugate • A combination of both a cevian triangle and an anticevian. • PA, PB, and PC are the points of the cevian triangle, P being the cevian triangle’s center point. These are also the points where the concurrent points, or QA, QB, and QC, join the original triangle. X is the point where the concurrent lines meet.

When not to use the formulas • Any concurrent lines inside or outside a triangle are cevians, but the formulas only work when the concurrent point falls inside the triangle. • They additionally only work when the end points of the cevians land on the lines of the triangle. • Therefore, because the cevians are altered, these formulas do not apply to anticevians or Ceva’s Conjugate

Review • AB’ . CA’ . BC’ B’C A’B C’A = 1, or AB’  CA’  BC’ = B’C  A’B  C’A • These formulas only apply when the concurrent point lies inside the triangle, and when the points of the cevians are on the line opposite the vertex it started from. • The definition of a cevian is very broad, and therefore concurrent lines previously taught, such as medians and altitudes, are cevians.

Aim: What is Ceva’s Theorem?