This article explains circle theorems, including tangents, sectors, angles and proofs (with thanks to Revision Maths).
Two Radii and a chord make an isosceles triangle.
Perpendicular Chord Bisection
The perpendicular from the centre of a circle to a chord will always bisect the chord (split it into two equal lengths).
Angles Subtended on the Same Arc
Angles formed from two points on the circumference are equal to other angles, in the same arc, formed from those two points.
Angle in a Semi-Circle
Angles formed by drawing lines from the ends of the diameter of a circle to its circumference form a right angle. So c is a right angle.
We can split the triangle in two by drawing a line from the centre of the circle to the point on the circumference our triangle touches.
We know that each of the lines which is a radius of the circle (the green lines) are the same length. Therefore each of the two triangles is isosceles and has a pair of equal angles.
But all of these angles together must add up to 180°, since they are the angles of the original big triangle.
Therefore x + y + x + y = 180, in other words 2(x + y) = 180. and so x + y = 90. But x + y is the size of the angle we wanted to find.
A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle- it just touches it).
A tangent to a circle forms a right angle with the circle’s radius, at the point of contact of the tangent.
Also, if two tangents are drawn on a circle and they cross, the lengths of the two tangents (from the point where they touch the circle to the point where they cross) will be the same.
Angle at the Centre
The angle formed at the centre of the circle by lines originating from two points on the circle’s circumference is double the angle formed on the circumference of the circle by lines originating from the same points. i.e. a = 2b.
You might have to be able to prove this fact:
OA = OX since both of these are equal to the radius of the circle. The triangle AOX is therefore isosceles and so ∠OXA = a Similarly, ∠OXB = b
Since the angles in a triangle add up to 180, we know that ∠XOA = 180 – 2a Similarly, ∠BOX = 180 – 2b Since the angles around a point add up to 360, we have that ∠AOB = 360 – ∠XOA – ∠BOX = 360 – (180 – 2a) – (180 – 2b) = 2a + 2b = 2(a + b) = 2 ∠AXB
Alternate Segment Theorem
This diagram shows the alternate segment theorem. In short, the red angles are equal to each other and the green angles are equal to each other.
You may have to be able to prove the alternate segment theorem:
We use facts about related angles
A tangent makes an angle of 90 degrees with the radius of a circle, so we know that ∠OAC + x = 90. The angle in a semi-circle is 90, so ∠BCA = 90. The angles in a triangle add up to 180, so ∠BCA + ∠OAC + y = 180 Therefore 90 + ∠OAC + y = 180 and so ∠OAC + y = 90 But OAC + x = 90, so ∠OAC + x = ∠OAC + y Hence x = y
A cyclic quadrilateral is a four-sided figure in a circle, with each vertex (corner) of the quadrilateral touching the circumference of the circle. The opposite angles of such a quadrilateral add up to 180 degrees.
Area of Sector and Arc Length
If the radius of the circle is r, Area of sector = πr2 × A/360 Arc length = 2πr × A/360
In other words, area of sector = area of circle × A/360 arc length = circumference of circle × A/360
Whether it’s the food I eat, the drinks I enjoy or the music I listen to, the main reason always seems to be a particular memory of a place, a person or a moment. This is a list of my favourite music with a few autobiographical grace notes…
Take A Chance On Me
My sister bought the single and I still remember I’m A Marionette on the B-side!
Alas We Aspire
Breathe (2 A.M.)
Waiting In Vain
Call me a softie, but Kate Beckinsale in Serendipity
Walking On Broken Glass
My dancing partner Vicki Kereszteny found this for rehearsing our cha
Stop The Rock
Claire Forlani at the foam party in Boys and Girls
Here Comes The Sun
Does anyone remember this as the theme to the Holiday programme…?!
Let It Be
I sang this to myself while cycling to every one of my finals
Eve, the Apple of My Eye
Take My Breath Away
Who can forget the Top Gun theme…?!
In My Life
From A Distance
This was the theme to Bosom Buddies, which gave us Tom Hanks and Donna Dixon!
Black Eyed Peas
I Gotta Feeling
To The End
Watching Blur at Mile End in the rain
Livin’ On A Prayer
Watching with Joe Oatley in Milton Keynes as he walked in front of 12′ screens full of feedback snow!
Boyz II Men
End Of The Road
Vicki gave me the Brit Awards CD from 1993, and this was on it…
Born In The USA
One of the best presents I ever had was the MTV Plugged CD given me by Mark Highman, and all these songs were on it…
I Wish I Were Blind
Thunder Road (MTV Plugged)
My Beautiful Reward
Oh Yeah (On The Radio)
More Than This
You Might Think
My first girlfriend Paula Buster from Canyon, TX
I have a soft spot for great theme tunes, and Carter Burwell pulled out all the stops for Conspiracy Theory…
Don’t Give Up
No Ordinary Morning
To Anne Crary, the only girl I ever asked to marry me, who introduced me to Chicane when I tried to put on the Bee Gees!
Moment In The Sun
Sunglasses At Night
Another one from my time in the States with Paula
Portuguese Love Theme
Love, Actually didn’t go down well with the Belgians next to me, but to me it was as if the director had exactly the same view of love (and good music!)
Glasgow Love Theme
PM’s Love Theme
Life On Mars?
The Goodbye Girl is one of the great (forgotten) romantic comedies. Richard Dreyfus and Marsha Mason at their best.
To my musical friend Eden James, who introduced me to David Gray in Brisbane when we ended up singing along to the guitar (and the odd possum on the roof!) after every house party…
My Oh My
Please Forgive Me
The Other Side
Here With Me
I have a bit of a weakness for US teen dramas, so I was easy meat for Shiri Appleby in Roswell
Romeo And Juliet
When I made the mistake of showing my diary to Elizabeth at Oxford, this is what I taped back-to-back and listened to over and over again to make the break-up hurt a bit less
So Far Away
Sultans Of Swing (Live Aid)
When I was away in Germany, I had to ask my mum to tape Live Aid, but this was the only song I liked (until I accidentally pressed the record button on my tape deck and had to do without it until Napster came along!)
Your Latest Trick
The Boys Of Summer
Emily Barker & The Red Clay Halo
Another great TV theme from The Shadow Line
Fats Waller & Benny Payne
After You’ve Gone
The other contender for the greatest present of all time. I told Claudine Kenny this was my favourite (unobtainable) jazz track, and she managed to track it down for my birthday.
Fine Young Cannibals
Club Can’t Handle Me
One For My Baby
One of the few songs to which I know all the words, which was embarrassing when my mum caught me singing it in Newton Abbot!
One of the happiest moments of my life came when we all trudged across the gravel on a perfect sunny October morning at Christ Church to have our matriculation photograph taken…and the second years played this from one of the windows in Peck! (They were fined £150…)
My sister caught sight of a list I was making of my favourite music and made me a couple of treasured mix tapes including this classic of urban disaffection
What Is Love
Another one from the Brit Awards CD from Vicki
My friend Tio made me a tape of ‘E-type jazz’ in exchange for a ride down the King’s Road in my 1965 carmine red Series 2 roadster, and this was my favourite
Hide And Seek
I heard this played over the funeral scene in The OC, so it was perhaps inevitable I should end up living there (Ormonde Court, not Orange County!)
Jon & Vangelis
I’ll Find My Way Home
I actually met Kasabian while playing the referee in Sky Sports’ last World Cup advert!
Thanks again to my sister Katharine for buying The Kick Inside. Did Kate Bush really write these songs when she was only 19?! Astonishing…
One of the best days of my life came when I took a day off to visit Singapore for a birthday treat for my ex-boss at Mercer. Champagne breakfast, waterskiing, sevens, a rickshaw ride, Singapore slings in Raffles, two beautiful girls and this song…
More thanks to Anne Crary for introducing me to this
Martin Solveig & Dragonette
Hello (Single Edit)
When We Collide
When I Look At You
O Magnum Mysterium
I heard this for the first time sitting next to Marina at a concert given by our friend Sam’s choir, and we were both blown away. What a perfect way to discover a wonderful piece!
The Great Gate of Kiev
I Love The Sound Of Breaking Glass
This memory is so far back, I don’t even know where it came from, but 1978 was a great year for new music (think Wuthering Heights!)
Turn Me On
The Nearness Of You
This is taken from the moment when Ed proposes to Carol in Ed (obviously!)
No More Lonely Nights
I’m not ashamed to say that Linda Kozlowski in a swimsuit had something to do with this…!
In Your Eyes
Cameron Fairchild, where are you now?! I met him in Amarillo, TX, and we were best friends for a few months during my year off. He was a fan of Peter Gabriel, and we played Risk, Axis & Allies and Conquest of Empire until the early hours before retiring to Denny’s for the all-day (and all-night!) breakfast…
Book Of Love
Peter Gabriel & Kate Bush
Don’t Give Up
Love You ‘Til The End
Jump (For My Love)
This is the theme from Cupid with Jeremy Piven and Paula Marshall, the best TV show never to be given a full season
O Mio Babbino Caro
I have to admit I heard this first on the soundtrack to A Room with a View, but they certainly picked the right version and the right soprano to sing it!
Chi Il Bel Sogno Di Doretta
Again from A Room with a View, and I almost prefer this aria now
I had a friend Pete Bouman at Oxford, who was an exchange student passionate about opera. One day in his rooms, he tried to play me an aria in which a soprano hit top C, but while he was fumbling with the fast forward and rewind buttons I heard this wonderful classic and made him play it for me. An early taste of Italia ’90!
Rhapsody on a Theme of Paganini
Groundhog Day is my favourite film ever, and this music has an important part to play!
2nd Piano Concerto, Mvt 1
Who but the most cynical of viewers could fail to be moved to tears by the score of Brief Encounter (even if my mother happened to catch me…!)?
2nd Piano Concerto, Mvt 2
2nd Piano Concerto, Mvt 3
What’d I Say
This one’s from Smallville. US teen dramas don’t get any cornier (so to speak…)
Start Me Up
Sympathy For The Devil
I was once at a party with Damian Hinds, who insisted on playing this even though nobody could dance to it!
Vicki again. Sigh…
Run DMC & Aerosmith
Walk This Way
This was the soundtrack to my very first race off the lights in Amarillo, TX. Eighty miles an hour, top down all the way. Glory…
Pop Looks Bach
Wind Of Change
Sixpence None The Richer
Sixpence None The Richer
There She Goes
House I Built
What a great Levi jeans advert! The music and pictures were in perfect harmony. Amish girl in black and white plus heartthrob and rock music – sounds like Witness!
From the soundtrack to Billy Elliot…
Back for Good (Radio Mix)
Dancing In The Moonlight
Anyone who went to Bananas in Val d’Isere around midnight would remember this one (and Angel by Robbie Williams)!
Drops Of Jupiter
Stand up, Craig Campbell. I only managed to make you dinner once that season in Mottaret, but we told so many stories while listening to so many songs that I have to credit you with the inspiration for this post!
Lie In The Sound
Trevor Jones & Randy Edelman
From The Last of the Mohicans. Definitely a better film than the book!
Trevor Jones & Randy Edelman
Last Of The Mohicans (Main Theme)
Where The Streets Have No Name
This is the theme to the promotional video of the Gnat Display Team, which I managed for a while in 2010-11
Who’s Gonna Ride Your Wild Horses
A Thousand Miles
Dammi Tu Forza, O Cielo!
Julia Roberts ‘peed her pants’ to this in Pretty Woman. What more can you say?!