Similarity (geometry)

In Euclidean geometry, two objects are similar if they both have the same shape, or one has the same shape as the mirror image of the other. More one can be obtained from the other by uniformly scaling with additional translation and reflection; this means that either object can be rescaled and reflected, so as to coincide with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other. For example, all circles are similar to each other, all squares are similar to each other, all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, isosceles triangles are not all similar to each other. If two angles of a triangle have measures equal to the measures of two angles of another triangle the triangles are similar. Corresponding sides of similar polygons are in proportion, corresponding angles of similar polygons have the same measure.

This article assumes that a scaling can have a scale factor of 1, so that all congruent shapes are similar, but some school textbooks exclude congruent triangles from their definition of similar triangles by insisting that the sizes must be different if the triangles are to qualify as similar. Two triangles, △ABC and △A′B′C′, are similar if and only if corresponding angles have the same measure: this implies that they are similar if and only if the lengths of corresponding sides are proportional, it can be shown that two triangles having congruent angles are similar, that is, the corresponding sides can be proved to be proportional. This is known as the AAA similarity theorem. Note that the "AAA" is a mnemonic: each one of the three A's refers to an "angle". Due to this theorem, several authors simplify the definition of similar triangles to only require that the corresponding three angles are congruent. There are several statements each of, necessary and sufficient for two triangles to be similar: The triangles have two congruent angles, which in Euclidean geometry implies that all their angles are congruent.

That is:If ∠BAC is equal in measure to ∠B′A′C′, ∠ABC is equal in measure to ∠A′B′C′ this implies that ∠ACB is equal in measure to ∠A′C′B′ and the triangles are similar. All the corresponding sides have lengths in the same ratio:AB/A′B′ = BC/B′C′ = AC/A′C′; this is equivalent to saying. Two sides have lengths in the same ratio, the angles included between these sides have the same measure. For instance:AB/A′B′ = BC/B′C′ and ∠ABC is equal in measure to ∠A′B′C′; this is known as the SAS similarity criterion. The "SAS" is a mnemonic: each one of the two S's refers to a "side"; when two triangles △ABC and △A′B′C′ are similar, one writes △ABC ∼ △A′B′C′. There are several elementary results concerning similar triangles in Euclidean geometry: Any two equilateral triangles are similar. Two triangles, both similar to a third triangle, are similar to each other. Corresponding altitudes of similar triangles have the same ratio as the corresponding sides. Two right triangles are similar if one other side have lengths in the same ratio.

Given a triangle △ABC and a line segment DE one can, with ruler and compass, find a point F such that △ABC ∼ △DEF. The statement that the point F satisfying this condition exists is Wallis's postulate and is logically equivalent to Euclid's parallel postulate. In hyperbolic geometry similar triangles are congruent. In the axiomatic treatment of Euclidean geometry given by G. D. Birkhoff the SAS similarity criterion given above was used to replace both Euclid's Parallel Postulate and the SAS axiom which enabled the dramatic shortening of Hilbert's axioms. Similar triangles provide the basis for many synthetic proofs in Euclidean geometry. Among the elementary results that can be proved this way are: the angle bisector theorem, the geometric mean theorem, Ceva's theorem, Menelaus's theorem and the Pythagorean theorem. Similar triangles provide the foundations for right triangle trigonometry; the concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence are proportional and corresponding angles taken in the same sequence are equal in measure.

However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles. Equality of all angles in sequence is not sufficient to guarantee similarity. A sufficient condition for similarity of polygons is that corresponding sides and diagonals are proportional. For given n, all regular n-gons are similar. Several types of curves have the property; these include: Circles Parabolas Hyperbolas of a specific eccentricity Ellipses of a specific eccentricity Catenaries Graphs of the logarithm function for different bases Graphs of the exponential function for different bases Logarithmic spirals are self-similar A similarity of a Euclidean space is a bijection f from the space onto itself that multiplies all distances by the same positive real number r, so that for any two points x and y we have d ( f, f (

Alessandro Cesarini

Alessandro Cesarini, bishop of Pistoia, was an Italian cardinal of the Roman Catholic Church. Born in Rome, the son of Agabito Cesarini, he became close to the Medici family Cardinal Giovanni di Lorenzo de' Medici, the future Pope Leo X, he received the deaconry of Sts. Sergius and Bacchus, opting for the deaconry of Santa Maria in Via Lata in 1523, he became known for his patronage of artists. He served as apostolic administrator of Pamplona, Spain from 1520 to 1538. In the sack of Rome by mutinous troops of Charles V in 1527, he was one of the cardinals held hostage, he participated in the conclave of 1521–1522, which elected Adrian VI. He became cardinal bishop and chose the suburbicarian see of Albano, Italy in 1540, he was appointed bishop of Palestrina, Italy in 1541, in which office he died February 13, 1542 in Rome. He was buried in his family’s tomb in the church of Santa Maria in Aracoeli in Rome

A. Doris Banks Henries

A. Doris Banks Henries was an American educator and writer in Liberia, Assistant Minister of Education during the Tolbert administration. Artiste Doris Banks was born in Live Oak and educated in Middletown, Connecticut, she received teacher training at Willimantic Normal School, pursued graduate education at Columbia University, where she earned a master's degree and completed her doctorate. During the 1930s, she worked as principal at Fuller Normal School in South Carolina. Dr. Banks first traveled to Liberia as a Methodist missionary in 1939, she became a professor at Liberia College in 1942. She was dean of William V. S. Tubman Teachers College until 1955, when she became an administrator at the University of Liberia. In 1959 she became Director of Higher Education and Textbook Research in Liberia's Department of Public Instruction. In that role, she worked for Africanization in curricular materials, saying "It should be the policy of African schools to include in all programs as much literature written by Africans as is available."

She served as president of the Liberian National Teachers Organization and the National YMCA, chaired the Liberian Methodist Board of Education. She rose to the rank of Assistant Minister of Education in 1978. Banks Henries wrote a biography of Liberian president William V. S. Tubman, published in 1967, she edited collections of Liberian poetry, of Liberian folklore. In her work as a government official, she was credited as author of many official reports and books about Liberia for young readers, she served as president of the Society of Liberian Authors. A. Doris Banks Henries; the Liberian Nation: A Short History. A. Doris Banks Henries. Civics for Liberian Schools. A. Doris Banks Henries. A Biography of William V. S. Tubman. A. Doris Banks Henries. Higher Education in Liberia: Retrospect — Present — Prospect. In 1942, Doris Banks married Richard Abrom Henries, a Liberian government official whose first wife was Angie Brooks, her husband, Speaker of the Liberian House of Representatives, was executed by firing squad in 1980, after a military coup.

Dr. Banks Henries died from cancer the following year, in Middletown, right after her 68th birthday. There is an A. Doris Banks Henries Scholarship Committee in Middletown, which grants scholarship funds to high school seniors in Middlesex County, Connecticut. A. Doris Banks was a member of Zeta Phi Beta sorority, she served as the first regional director for Africa in 1949. She was a member of the Delta Iota Zeta chapter in Monrovia, Africa